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Zubov’s Method for Differential Games
Lars Grune
Mathematisches InstitutUniversitat Bayreuth
Joint work with Oana Silvia Serea,Ecole Polytechnique, Palaiseau, France
International Workshop “The Dynamics of Control”Irsee, 1st–3rd October, 2010
Happy Birthday Fritz!
Zubov’s Method for Differential Games
Lars Grune
Mathematisches InstitutUniversitat Bayreuth
Joint work with Oana Silvia Serea,Ecole Polytechnique, Palaiseau, France
International Workshop “The Dynamics of Control”Irsee, 1st–3rd October, 2010
Happy Birthday Fritz!
Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd
(without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd (without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd (without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd (without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
Example for a Domain of AttractionFluid Dynamics: Explanation of the difference between linearstability and experimental instability for large Reynoldsnumbers [Trefethen et al., Science, 1993]
Lars Grune, Zubov’s Method for Differential Games, p. 3 of 23
Zubov’s Equation [1964]For a continuous function h : Rd → R≥0 withh(x) = 0⇔ x = x∗ consider the PDE “Zubov’s Equation”
Dw(x) · f(x) = −h(x)(1− w(x))
with w : Rd → R and boundary condition w(x∗) = 0
Then: under suitable conditions on h this equation has aunique solution w : Rd → [0, 1] with
w(x) = 0 ⇔ x = x∗
and D satisfies the level set characterization
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Lars Grune, Zubov’s Method for Differential Games, p. 4 of 23
Zubov’s Equation [1964]For a continuous function h : Rd → R≥0 withh(x) = 0⇔ x = x∗ consider the PDE “Zubov’s Equation”
Dw(x) · f(x) = −h(x)(1− w(x))
with w : Rd → R and boundary condition w(x∗) = 0
Then: under suitable conditions on h this equation has aunique solution w : Rd → [0, 1] with
w(x) = 0 ⇔ x = x∗
and D satisfies the level set characterization
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Lars Grune, Zubov’s Method for Differential Games, p. 4 of 23
Example
x1(t) = −x1(t) + x1(t)3, x2(t) = −x2(t) + x2(t)3
D = [−1, 1]2
, h(x) = 5‖x‖2
−10
1
−1
0
1
0
0.2
0.4
0.6
0.8
1
x1
x2
v
Lars Grune, Zubov’s Method for Differential Games, p. 5 of 23
Example
x1(t) = −x1(t) + x1(t)3, x2(t) = −x2(t) + x2(t)3
D = [−1, 1]2, h(x) = 5‖x‖2
−10
1
−1
0
1
0
0.2
0.4
0.6
0.8
1
x1
x2
v
Lars Grune, Zubov’s Method for Differential Games, p. 5 of 23
Integral Equation
Dw(x) · f(x) = −h(x)(1− w(x))
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Why does this characterization hold?
Integration of Zubov’s equation and subsequentintegration by parts yields the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
Thus:
Φ(t, x)→ x∗ ⇔∫ ∞
0
h(Φ(t, x))dt <∞ ⇔ w(x) < 1
Lars Grune, Zubov’s Method for Differential Games, p. 6 of 23
Integral Equation
Dw(x) · f(x) = −h(x)(1− w(x))
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Why does this characterization hold?
Integration of Zubov’s equation and subsequentintegration by parts yields the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
Thus:
Φ(t, x)→ x∗ ⇔∫ ∞
0
h(Φ(t, x))dt <∞ ⇔ w(x) < 1
Lars Grune, Zubov’s Method for Differential Games, p. 6 of 23
Integral Equation
Dw(x) · f(x) = −h(x)(1− w(x))
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Why does this characterization hold?
Integration of Zubov’s equation and subsequentintegration by parts yields the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
Thus:
Φ(t, x)→ x∗ ⇔∫ ∞
0
h(Φ(t, x))dt <∞ ⇔ w(x) < 1
Lars Grune, Zubov’s Method for Differential Games, p. 6 of 23
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction D
an existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
Control and PerturbationIn this talk we consider generalizations of this method forcontrolled and deterministically perturbed systems
x(t) = f(x(t), u(t), v(t))
with x(t) ∈ Rd
u ∈ U = {u : [0,∞)→ U, measurable}v ∈ V = {v : [0,∞)→ V, measurable}U ⊂ Rm, V ⊂ Rl compact
Problem: stabilization under uncertainty
u = control, trying to achieve Φ(t, x0, u, v)→ x∗
v = perturbation, trying to keep Φ(t, x0, u, v) away from x∗
(convergence to x∗ = 0 can be generalized to arbitrary compact sets)
Lars Grune, Zubov’s Method for Differential Games, p. 8 of 23
Control and PerturbationIn this talk we consider generalizations of this method forcontrolled and deterministically perturbed systems
x(t) = f(x(t), u(t), v(t))
with x(t) ∈ Rd
u ∈ U = {u : [0,∞)→ U, measurable}v ∈ V = {v : [0,∞)→ V, measurable}U ⊂ Rm, V ⊂ Rl compact
Problem: stabilization under uncertainty
u = control, trying to achieve Φ(t, x0, u, v)→ x∗
v = perturbation, trying to keep Φ(t, x0, u, v) away from x∗
(convergence to x∗ = 0 can be generalized to arbitrary compact sets)
Lars Grune, Zubov’s Method for Differential Games, p. 8 of 23
Control and PerturbationIn this talk we consider generalizations of this method forcontrolled and deterministically perturbed systems
x(t) = f(x(t), u(t), v(t))
with x(t) ∈ Rd
u ∈ U = {u : [0,∞)→ U, measurable}v ∈ V = {v : [0,∞)→ V, measurable}U ⊂ Rm, V ⊂ Rl compact
Problem: stabilization under uncertainty
u = control, trying to achieve Φ(t, x0, u, v)→ x∗
v = perturbation, trying to keep Φ(t, x0, u, v) away from x∗
(convergence to x∗ = 0 can be generalized to arbitrary compact sets)
Lars Grune, Zubov’s Method for Differential Games, p. 8 of 23
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J
zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
Information Exchange between u and vWhat do u and v know about each other?
Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)
General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖
(can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
Formal Derivation of Zubov’s Equation
w+(x) := supβ∈∆
infu∈U
{1− e−
R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
}satisfies for all T > 0 the optimality principle
w+(x) = supβ∈∆
infu∈U
{1− e−
R T0 h(Φ(t,x,u,v),u(t),v(t))dt[1− w+(Φ(T, x, u, v))]
}
Division by −T and passing to the limit for T → 0 yields theHamilton–Jacobi–Isaacs equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
“generalized Zubov Equation”
Lars Grune, Zubov’s Method for Differential Games, p. 14 of 23
Formal Derivation of Zubov’s Equation
w+(x) := supβ∈∆
infu∈U
{1− e−
R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
}satisfies for all T > 0 the optimality principle
w+(x) = supβ∈∆
infu∈U
{1− e−
R T0 h(Φ(t,x,u,v),u(t),v(t))dt[1− w+(Φ(T, x, u, v))]
}Division by −T and passing to the limit for T → 0 yields theHamilton–Jacobi–Isaacs equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
“generalized Zubov Equation”
Lars Grune, Zubov’s Method for Differential Games, p. 14 of 23
Formal Derivation of Zubov’s Equation
w+(x) := supβ∈∆
infu∈U
{1− e−
R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
}satisfies for all T > 0 the optimality principle
w+(x) = supβ∈∆
infu∈U
{1− e−
R T0 h(Φ(t,x,u,v),u(t),v(t))dt[1− w+(Φ(T, x, u, v))]
}Division by −T and passing to the limit for T → 0 yields theHamilton–Jacobi–Isaacs equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
“generalized Zubov Equation”
Lars Grune, Zubov’s Method for Differential Games, p. 14 of 23
Formal Derivation of Zubov’s Equationw+ formally satisfies the generalized Zubov equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
Likewise, w− formally satisfies the generalized Zubov equation
H−(x,w−(x), Dw−(x)) = 0
with Hamiltonian
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 15 of 23
Formal Derivation of Zubov’s Equationw+ formally satisfies the generalized Zubov equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
Likewise, w− formally satisfies the generalized Zubov equation
H−(x,w−(x), Dw−(x)) = 0
with Hamiltonian
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 15 of 23
Nonsmooth SolutionProblem: w+ and w− from the integral equations are typicallynonsmooth
Example: w+ for x(t) = −x(t) + u(t)v(t)x(t)3, withU = V = {−1, 1}, h(x, u, v) = x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
viscositysolution
Lars Grune, Zubov’s Method for Differential Games, p. 16 of 23
Nonsmooth SolutionProblem: w+ and w− from the integral equations are typicallynonsmooth
Example: w+ for x(t) = −x(t) + u(t)v(t)x(t)3, withU = V = {−1, 1}, h(x, u, v) = x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
viscositysolution
Lars Grune, Zubov’s Method for Differential Games, p. 16 of 23
Nonsmooth SolutionProblem: w+ and w− from the integral equations are typicallynonsmooth
Example: w+ for x(t) = −x(t) + u(t)v(t)x(t)3, withU = V = {−1, 1}, h(x, u, v) = x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
viscositysolution
Lars Grune, Zubov’s Method for Differential Games, p. 16 of 23
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
Existence and UniquenessWith this solution concept and with the help of “sub– andsuperoptimality principles” for viscosity super– andsubsolutions [Soravia 95] we arrive at the following Theorem:
w+ is the unique continuous viscosity solution of the equation
supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w+(0) = 0
w− is the unique continuous viscosity solution of the equation
infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w−(0) = 0
Furthermore, the characterizations D+ = (w+)−1([0, 1)) andD− = (w−)−1([0, 1)) hold
Lars Grune, Zubov’s Method for Differential Games, p. 18 of 23
Existence and UniquenessWith this solution concept and with the help of “sub– andsuperoptimality principles” for viscosity super– andsubsolutions [Soravia 95] we arrive at the following Theorem:
w+ is the unique continuous viscosity solution of the equation
supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w+(0) = 0
w− is the unique continuous viscosity solution of the equation
infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w−(0) = 0
Furthermore, the characterizations D+ = (w+)−1([0, 1)) andD− = (w−)−1([0, 1)) hold
Lars Grune, Zubov’s Method for Differential Games, p. 18 of 23
Existence and UniquenessWith this solution concept and with the help of “sub– andsuperoptimality principles” for viscosity super– andsubsolutions [Soravia 95] we arrive at the following Theorem:
w+ is the unique continuous viscosity solution of the equation
supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w+(0) = 0
w− is the unique continuous viscosity solution of the equation
infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w−(0) = 0
Furthermore, the characterizations D+ = (w+)−1([0, 1)) andD− = (w−)−1([0, 1)) hold
Lars Grune, Zubov’s Method for Differential Games, p. 18 of 23
ExampleConsider the example from before
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
For h(x, u, v) = x2 we can compute explicitly
w+(x) =
{1−√
1− x2, |x| < 11, |x| ≥ 1
w−(x) =
√1 + x2 − 1√
1 + x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
W−(x
)
This confirms D+ = (−1, 1) and D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 19 of 23
ExampleConsider the example from before
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
For h(x, u, v) = x2 we can compute explicitly
w+(x) =
{1−√
1− x2, |x| < 11, |x| ≥ 1
w−(x) =
√1 + x2 − 1√
1 + x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
W−(x
)
This confirms D+ = (−1, 1) and D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 19 of 23
ExampleConsider the example from before
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
For h(x, u, v) = x2 we can compute explicitly
w+(x) =
{1−√
1− x2, |x| < 11, |x| ≥ 1
w−(x) =
√1 + x2 − 1√
1 + x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
W−(x
)
This confirms D+ = (−1, 1) and D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 19 of 23
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds.
Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
When does D+ = D− hold?
For the special case of h(x, u, v) = h(x) we get
H+(x,w, p) = H−(x,w, p)
⇔ supu∈U
infv∈V{−p · f(x, u, v)} = inf
v∈Vsupu∈U{−p · f(x, u, v)}
This condition (for all p ∈ Rn) is known as Isaacs’ condition
Theorem: Isaacs’ condition implies D+ = D−
This theorem extends a well known result from capture basinsin finite time pursuit evasion games to domains ofcontrollability of asymptotically controllable sets
Lars Grune, Zubov’s Method for Differential Games, p. 21 of 23
When does D+ = D− hold?
For the special case of h(x, u, v) = h(x) we get
H+(x,w, p) = H−(x,w, p)
⇔ supu∈U
infv∈V{−p · f(x, u, v)} = inf
v∈Vsupu∈U{−p · f(x, u, v)}
This condition (for all p ∈ Rn) is known as Isaacs’ condition
Theorem: Isaacs’ condition implies D+ = D−
This theorem extends a well known result from capture basinsin finite time pursuit evasion games to domains ofcontrollability of asymptotically controllable sets
Lars Grune, Zubov’s Method for Differential Games, p. 21 of 23
When does D+ = D− hold?
For the special case of h(x, u, v) = h(x) we get
H+(x,w, p) = H−(x,w, p)
⇔ supu∈U
infv∈V{−p · f(x, u, v)} = inf
v∈Vsupu∈U{−p · f(x, u, v)}
This condition (for all p ∈ Rn) is known as Isaacs’ condition
Theorem: Isaacs’ condition implies D+ = D−
This theorem extends a well known result from capture basinsin finite time pursuit evasion games to domains ofcontrollability of asymptotically controllable sets
Lars Grune, Zubov’s Method for Differential Games, p. 21 of 23
ExampleIn our example
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}we have D+ = (−1, 1) 6= (−∞,∞) = D−
Isaacs’ condition must be violated
Indeed, for p = 1 and x = 1 we have
p · f(x, u, v) = −1 + uv
and thus
supu∈U
infv∈V{−p · f(x, u, v)} = sup
u∈Uinfv∈V{1− uv} = 0
but
infv∈V
supu∈U{−p · f(x, u, v)} = inf
v∈Vsupu∈U{1− uv} = 2
Lars Grune, Zubov’s Method for Differential Games, p. 22 of 23
ExampleIn our example
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}we have D+ = (−1, 1) 6= (−∞,∞) = D−
Isaacs’ condition must be violated
Indeed, for p = 1 and x = 1 we have
p · f(x, u, v) = −1 + uv
and thus
supu∈U
infv∈V{−p · f(x, u, v)} = sup
u∈Uinfv∈V{1− uv} = 0
but
infv∈V
supu∈U{−p · f(x, u, v)} = inf
v∈Vsupu∈U{1− uv} = 2
Lars Grune, Zubov’s Method for Differential Games, p. 22 of 23
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23