Martino Bardi

Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations

Preface.- Basic notations.- Outline of the main ideas on a model problem.- Continuous viscosity solutions of Hamilton-Jacobi equations.- Optimal control problems with continuous value functions: unrestricted state space.- Optimal control problems with continuous value functions: restricted state space.- Discontinuous viscosity solutions and applications.- Approximation and perturbation problems.- Asymptotic problems.- Differential Games.- Numerical solution of Dynamic Programming.- Nonlinear H-infinity control by Pierpaolo Soravia.- Bibliography.- Index

An approximation scheme for the minimum time function

This paper presents an approximation scheme for the nonlinear minimum time problem with compact target. The scheme is derived from a discrete dynamic programming principle and the main convergence result is obtained by applying techniques related to discontinuous viscosity solutions for Hamilton–Jacobi equations. The convergence is proved under general controllability assumptions on both the continuous-time and the discrete-time systems. An explicit sufficient condition on the system and the target ensuring the desired controllability is given. This condition is shown to be necessary and sufficient for the Lipschitz continuity of the minimum time function if the target is smooth. An extension to the case of a point-shaped target is given.

EXPLICIT SOLUTIONS OF SOME LINEAR-QUADRATIC MEAN FIELD GAMES

We consider

Numerical Methods for Pursuit-Evasion Games via Viscosity Solutions

N

Viscosity Solutions Methods for Singular Perturbations in Deterministic and Stochastic Control

-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number

A boundary value problem for the minimum-time function

N

The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations

of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [22]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.

Fully Discrete Schemes for the Value Function of Pursuit-Evasion Games

We present a class of numerical schemes for the Isaacs equation of pursuit-evasion games. We consider continuous value functions, where the solution is interpreted in the viscosity sense, as well as discontinuous value functions, where the notion of viscosity envelope-solution is needed. The convergence of the approximation scheme to the value function of the game is proved in both cases. A priori estimates of the convergence in L∞ are established when the value function is Holder continuous. We also treat problems with state constraints and discuss several issues concerning the implementation of the approximation scheme, the synthesis of approximate feedback controls, and the approximation of optimal trajectories. The efficiency of the algorithm is illustrated by a number of numerical tests, either in the case of one player (i.e., minimum time problem) or for some 2-players games.

Linear-Quadratic N-person and Mean-Field Games with Ergodic Cost

Viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order, degenerate parabolic, Hamilton--Jacobi--Bellman equations. This characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions. The results apply to cases where the usual order reduction method does not give the correct limit, and to systems with fast state variables depending nonlinearly on the control. Some connections with ergodic control and periodic homogenization are discussed.

Convergence of Discrete Schemes for Discontinuous Value Functions of Pursuit-Evasion Games

A natural boundary value problem for the dynamic programming partial differential equation associated with the minimum time problem is proposed. The minimum time function is shown to be the unique viscosity solution of this boundary value problem.