Nonlinear elliptic and parabolic equations of the second order

(2000). Lp- Theory for fully nonlinear uniformly parabolic equations. Communications in Partial Differential Equations: Vol. 25, No. 11-12, pp. 1997-2053.

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Lp- Theory for fully nonlinear uniformly parabolic equations

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.

Numerical Methods for Pursuit-Evasion Games via Viscosity Solutions

This paper deals with the optimal control of a stochastic delay differential equation arising in the management of a pension fund with surplus. The problem is approached by the tool of a representation in infinite dimension. We show the equivalence between the one-dimensional delay problem and the associated infinite-dimensional problem without delay. Then we prove that the value function is continuous in this infinite-dimensional setting. These results represent a starting point for the investigation of the associated infinite-dimensional Hamilton–Jacobi–Bellman equation in the viscosity sense and for approaching the problem by numerical algorithms. Also an example with complete solution of a simpler but similar problem is provided.

A stochastic control problem with delay arising in a pension fund model

We specify a dynamic model in which agents adjust their decisions toward higher payoffs, subject to normal error. This process generates a probability distribution of players' decisions that evolves over time according to the Fokker-Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This "logit equilibrium" forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.

Noisy Directional Learning and the Logit Equilibrium

We consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions, where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous. We prove that the value function is a viscosity solution of the Bellman equation and is in fact the minimal nonnegative supersolution.

A Viscosity Approach to Infinite-Dimensional Hamilton--Jacobi Equations Arising in Optimal Control with State Constraints

On Differential Games for Infinite-Dimensional Systems with Nonlinear, Unbounded Operators

Lyapunov Functions for Infinite-Dimensional Systems

This paper studies the ${\cal H}_\infty$ control problem for a nonlinear, unbounded, infinite dimensional system with state constraints. We characterize the solvability of the problem by means of a Hamilton--Jacobi--Isaacs (HJI) equation, proving that the ${\cal H}_\infty$ problem can be solved if and only if the HJI equation has a positive definite viscosity supersolution, vanishing and continuous at the origin. In order to do so, the standard definition of the ${\cal H}_\infty$ problem has to be relaxed by using the theory of differential games. We apply our results to the one phase Stefan problem.

Differential Games and Nonlinear \boldmath$\cal H_\infty$ Control in Infinite Dimensions

The generalized Stepanov theorem is derived from the Alexandrov theorem on the twice differentiability of convex functions. A parabolic version of the generalized Stepanov theorem is also proved. In the first part of this note we provide a new proof of the generalized Stepanov theorem. This classical result is due to Calderón and Zygmund [3] (see also Oliver [12]), but is usually associated with Stepanov’s name because it generalizes the Stepanov theorem (see e.g. [6]). The result we prove below (Theorem 1) constitutes a special case of a general theorem in [3]. Recently this particular version found applications in proving the twice differentiability a.e. of viscosity solutions of elliptic partial differential equations, see [11], [14] and [2]. The only complete proof of the generalized Stepanov theorem the authors are aware of is contained in [3], where Whitney’s extension theorem is used. In this note the generalized Stepanov theorem will be proved by means of the Aleksandrov theorem on the twice differentiability of convex functions [1]; see also [5], [9], [10] or the appendix in [4] for more modern treatments. In the second part of this note we show how to modify our proof to obtain a parabolic version of the generalized Stepanov theorem (Theorem 3). A result of this type is needed to prove the differentiability a.e. twice in x and once in t of viscosity solutions of parabolic equations. To the best of the authors’ knowledge this result is original, though some relevant arguments appear in [16]. | · | and 〈·, ·〉 will stand for the Euclidean norm and inner-product in R, and Br(x) will denote the open ball in R of radius r centered at x. Given a measurable set A in an Euclidean space, |A| will denote its Lebesgue measure. Recall some notation from [3] (see also [17]). Let u : Ω → R, Ω ⊂ R, be bounded and x ∈ Ω. We say that u ∈ T 2 ∞(x) (u ∈ t∞(x), resp.) if there exists an affine function Px (a quadratic function Qx) such that sup y∈Br(x)∩Ω |u(y)− Px(y)| ≤ O(r) ( sup y∈Br(x)∩Ω |u(y)−Qx(y)| ≤ o(r) as r ↓ 0, resp. ) . Observe that u ∈ t∞(x) if and only if u possesses a second order Taylor series expansion at x whose remainder behaves like o(r). If this is the case we will say Received by the editors February 21, 1996. 1991 Mathematics Subject Classification. Primary 26B05. This work was supported by the Australian Research Council. c ©1997 American Mathematical Society

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Maciej Kocan

Papers

A Viscosity Approach to Infinite-Dimensional Hamilton--Jacobi Equations Arising in Optimal Control with State Constraints

On Differential Games for Infinite-Dimensional Systems with Nonlinear, Unbounded Operators

Lyapunov Functions for Infinite-Dimensional Systems

Differential Games and Nonlinear \boldmath$\cal H_\infty$ Control in Infinite Dimensions

On the generalized Stepanov theorem