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A Differential Game Described by a Hyperbolic System

Souroujon, Diko (1999)

Serdica Mathematical Journal

An antagonistic differential game of hyperbolic type with a separable linear vector pay-off function is considered. The main result is the description of all ε-Slater saddle points consisting of program strategies, program ε-Slater maximins and minimaxes for each ε ∈ R^N > for this game. To this purpose, the considered differential game is reduced to find the optimal program strategies of two multicriterial problems of hyperbolic type. The application...

Almost sure properties of controlled diffusions and worst case properties of deterministic systems

Martino Bardi, Annalisa Cesaroni (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We compare a general controlled diffusion process with a deterministic system where a second controller drives the disturbance against the first controller. We show that the two models are equivalent with respect to two properties: the viability (or controlled invariance, or weak invariance) of closed smooth sets, and the existence of a smooth control Lyapunov function ensuring the stabilizability of the system at an equilibrium.


Boundary-influenced robust controls: two network examples

Martin V. Day (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the differential game associated with robust control of a system in a compact state domain, using Skorokhod dynamics on the boundary. A specific class of problems motivated by queueing network control is considered. A constructive approach to the Hamilton-Jacobi-Isaacs equation is developed which is based on an appropriate family of extremals, including boundary extremals for which the Skorokhod dynamics are active. A number of technical lemmas and a structured verification theorem...

Conflict-Controlled Processes Involving Fractional Differential Equations with Impulses

Matychyn, Ivan, Chikrii, Arkadii, Onyshchenko, Viktoriia (2012)

Mathematica Balkanica New Series

MSC 2010: 34A08, 34A37, 49N70Here we investigate a problem of approaching terminal (target) set by a system of impulse differential equations of fractional order in the sense of Caputo. The system is under control of two players pursuing opposite goals. The first player tries to bring the trajectory of the system to the terminal set in the shortest time, whereas the second player tries to maximally put off the instant when the trajectory hits the set, or even avoid this meeting at all. We derive...

Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem

Claudio Marchi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space  [0, +∞) × ℝn and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.

Deterministic minimax impulse control in finite horizon: the viscosity solution approach

Brahim El Asri (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We study here the impulse control minimax problem. We allow the cost functionals and dynamics to be unbounded and hence the value functions can possibly be unbounded. We prove that the value function of the problem is continuous. Moreover, the value function is characterized as the unique viscosity solution of an Isaacs quasi-variational inequality. This problem is in relation with an application in mathematical finance.

Differential games of partial information forward-backward doubly SDE and applications

Eddie C. M. Hui, Hua Xiao (2014)

ESAIM: Control, Optimisation and Calculus of Variations

This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle...

Dynamic Programming Principle for tug-of-war games with noise

Juan J. Manfredi, Mikko Parviainen, Julio D. Rossi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that...

Dynamic Programming Principle for tug-of-war games with noise

Juan J. Manfredi, Mikko Parviainen, Julio D. Rossi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that the value functions of this game satisfy the Dynamic Programming Principle...

Dynamic Programming Principle for tug-of-war games with noise

Juan J. Manfredi, Mikko Parviainen, Julio D. Rossi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that...

Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls

Hong Qiu, Jiongmin Yong (2013)

ESAIM: Control, Optimisation and Calculus of Variations

A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls,...

Linear-quadratic differential games: from finite to infinite dimension

Michel C. Delfour (2008)

Applicationes Mathematicae

The object of this paper is the generalization of the pioneering work of P. Bernhard [J. Optim. Theory Appl. 27 (1979)] on two-person zero-sum games with a quadratic utility function and linear dynamics. It relaxes the semidefinite positivity assumption on the matrices in front of the state in the utility function and introduces affine feedback strategies that are not necessarily L²-integrable in time. It provides a broad conceptual review of recent results in the finite-dimensional case for which...

Nash equilibrium payoffs for stochastic differential games with reflection

Qian Lin (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.

Objective function design for robust optimality of linear control under state-constraints and uncertainty

Fabio Bagagiolo, Dario Bauso (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.

Objective function design for robust optimality of linear control under state-constraints and uncertainty

Fabio Bagagiolo, Dario Bauso (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.

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