Uncertain saddle point equilibrium differential games with non-anticipating strategies

doi:10.1016/j.ejcon.2018.01.004

European Journal of Control

Volume 41, May 2018, Pages 8-15

https://doi.org/10.1016/j.ejcon.2018.01.004 Get rights and content

Abstract

In this paper, we investigate uncertain saddle point equilibrium differential games under uncertain environment. We propose an optimistic value game model and define the value function of the game by introducing the concept of non-anticipating strategy. We prove the continuity and dynamic programming property of the value function. Then we derive the uncertain Hamilton–Jacobi–Isaacs equation by the viscosity solution approach.

Introduction

In the past few decades, game theory has been an active research field in operations research and control theory. Von Meumann and Morgenstern [24] first established the modern game theory. Later, Isaacs [15] studied a two person zero sum differential game model in a dynamical system which initiated the research of differential game theory. Pontryagin [25] considered a class of differential games with the maximum principle theorem. Friedman [12] and Berkovitz [2] investigated the existence theorem and approximation method for differential games.

In previous work, one important way to solve the two person zero sum differential game is to transform the original problem to solving a PDE (called HJI equaiton). An important assumption is that the value function of the game is assumed to be sufficiently smooth (e.g. twice differentiable) to make sense of the related HJI equation. Nevertheless, this assumption is usually impossible to be achieved. Many researchers [5], [9], [11] has worked on this difficulty with some relaxed assumptions. The breakthrough is the establishment of concept of non-anticipating strategy and viscosity solution (see e.g. [6], [8], [10], [18]).

Stochastic differential game [1], [3], [7], [14], [28] also received much attention. However, noises in some particular dynamical systems do not behave like randomness, such as the price of new stock, bridge strength and oil field reserves. There are no enough samples to ensure the estimated probability distribution of the noises is close enough to the long-run cumulative frequency. Hence, stochastic differential equations are not able to appropriately model these dynamical systems [23]. To estimate this kind of indeterministic noises, people have to invite some domain experts to evaluate the belief degree that each event may occur. Liu [19] founded uncertainty theory in 2007 to rationally deal with personal belief degrees. Nowadays, uncertain theory has been applied to many fields (see e.g. [4], [26], [27], [30], [34], [35]). Thus, for those dynamical systems which cannot be appropriately described by stochastic differential equations, we may use uncertain differential equations. Two person zero sum uncertain differential games were analyzed (see e.g. [13], [29], [31], [32]).

In this paper, we consider a two person zero sum uncertain differential game with non-anticipating strategies. The rest of this paper is organized as follows. In Section 2, we review some basic concepts about uncertainty theory. In Section 3, we formulate our saddle point equilibrium game model and introduce the non-anticipating strategy. In Section 4, we discuss the properties of the value function. In the Section 5, we establish the relationship between the value function and the uncertain HJI equation with the viscosity solution. In the last section, we give an example to illustrate our results.

Section snippets

Preliminaries

In this section, we introduce some important basic concepts about uncertainty theory, which are used throughout the paper.

Uncertainty theory is a branch of axiomatic mathematics to deal with human uncertainty arising from the belief degrees. Let Γ be a nonempty set and $L$ be a σ-algebra over Γ. Each element $Λ \in L$ is called an event. Then uncertain measure $M {Λ}$ is used to evaluate the belief degree that each event Λ may occur. The axiomatic definition is as follows.

Definition 1

[19] A set function $M$ defined on

Problem formulation

In this section, we formulate our uncertain saddle point equilibrium differential game model and provide some estimates about the system states. In addition, we introduce the notion of non-anticipating strategy.

We consider a control system described by an uncertain differential equation as follows: $\begin{matrix} {\begin{matrix} d X_{s} = f (s, X_{s}, u_{1}, u_{2}) d s + g (s, X_{s}, u_{1}, u_{2}) d C_{s}, t \leq s \leq T, \\ X_{t} = x, \end{matrix} \end{matrix}$ where $f : [0, T] \times R^{n} \times U_{1} \times U_{2} \to R^{n},$ $g : [0, T] \times R^{n} \times U_{1} \times U_{2} \to R^{n \times k},$ with $U_{1} \in R^{p}, U_{2} \in R^{p}$ being some non-empty closed convex sets. In the above system, X_s is the state

Properties of the Elliott–Kalton lower and upper value

In this section, we discuss some basic properties of the Elliott–Kalton upper and lower value functions. First, we provide some preparations.

Lemma 2

Let ξ and η be two uncertain variables, and α ∈ (0, 1]. If ξ(γ) ≤ η(γ), ∀γ ∈ Γ, we have $ξ_{\sup} (α) \leq η_{\sup} (α)$ .

Proof

Since ξ(γ) ≤ η(γ) for any γ ∈ Γ, we have ${ξ (γ) \geq r} \subset {η (γ) \geq r}, \forall r \in R .$ Thus, we have $M {ξ (γ) \geq r} \leq M {η (γ) \geq r},$ which yields $\begin{matrix} {M {ξ (γ) \geq r} \geq α} \subset {M {η (γ) \geq r} \geq α} . \end{matrix}$ Then we know that $\sup {r | M {ξ (γ) \geq r} \geq α} \leq \sup {r | M {η (γ) \geq r} \geq α},$ i.e. $ξ_{\sup} (α) \leq η_{\sup} (α)$ . □

Lemma 3

Let C_t be a Liu process. And K

Viscosity solutions to uncertain Hamilton–Jacobi–Isaacs equations

In this section, we prove that the Elliott–Kalton upper and lower value functions are unique viscosity solutions to uncertain Hamilton–Jacobi–Isaacs equations. First, we introduce the concept of viscosity solution.

Definition 7

Assume $H : [0, T] \times R^{n} \times R^{n} \to R$ is continuous. Also $h : R^{n} \to R$ is assumed to be continuous. A continuous function J(t, x) is called a viscosity solution to the following Hamilton–Jacobi equation $\begin{matrix} J_{t} (t, x) + H (t, x, \nabla_{x} J (t, x)) = 0, (t, x) \in [0, T] \times R^{n} \end{matrix}$ provided that Eq. (10) holds with $J (T, x) = h (x),$ and for any

An example

Now, we provide an example to illustrate the significance of the viscosity solution approach for the uncertain saddle point differential game. HJI equation is an important indirect way to study the property of value function of a game. However the HJI equation may have no classical solution.

Consider an uncertain system: $\begin{matrix} {\begin{matrix} d X_{s} = (u_{1} - u_{2}) X_{s} d s + σ X_{s} d C_{s}, s \in [0, 1] \\ X_{0} = x_{0}, \end{matrix} \end{matrix}$ where the admissible set is $U_{i} [0, 1] = {u_{i} : [0, 1] \to [- 1, 1] | u_{i} is measurable}$ . And the cost functional is given by $\begin{matrix} V (0, x_{0}, u_{1}, u_{2}) = {(X_{1})}_{\sup} (α) . \end{matrix}$ For any $(t$

Conclusion

In this paper, we introduced the concept of non-anticipating strategy with which we define the Elliott–Kalton value function for uncertain saddle point equilibrium differential game. The continuity of this value function was investigated. Furthermore, we proved that the Elliott–Kalton value function satisfies the dynamic programming condition. Finally, we utilized the viscosity solution theory to characterize the value function with an uncertain Hamilton–Jacobi–Isaacs equation.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61673011).

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View full text

Uncertain saddle point equilibrium differential games with non-anticipating strategies

Abstract

Introduction

Section snippets

Preliminaries

Problem formulation

Properties of the Elliott–Kalton lower and upper value

Viscosity solutions to uncertain Hamilton–Jacobi–Isaacs equations

An example

Conclusion

Acknowledgments

J. Differ. Equ.

Sys. Control Lett.

Eur. J. Control

On a class of linear stochastic differential games

IEEE Trans. Automat. Contr.

The existence of value and saddle point in games of fixed duration

SIAM J. Control Optim.

Stochastic differential games and viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations

SIAM J. Control Optim.

Indefinite LQ optimal control with equality constraint for discrete-time uncertain systems

Jpn. J. Ind. Appl. Math.

Hamilton’s theory and generalized solutions of the Hamilton–Jacobi equation

J. Math. Mech.

Viscosity solutions of Hamilton–Jacobi equations

Trans. Am. Math. Soc.

The existence of value in stochastic differential games

SIAM J. Control Optim.

The existence of value in differential games

Mem. Am. Math. Soc.

On solving certain nonlinear partial differential equations by accretive operator methods

Israel J. Math.

Differential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equations

Indiana U. Math. J.

Differential Games

Uncertain Shapley value of coalitional game with application to supply chain alliance

Appl. Soft Comput.

Differential Games

Uniqueness of unbounded viscosity solutions of Hamilton–Jacobi equations

Indiana U. Math. J.