We study the asymptotic behavior of $\lambda v_{\lambda }$ as $\lambda \to 0^{+}$, where $v_{\lambda }$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case)$$\lambda v_{\lambda }+H(x,D{v}_{\lambda })=0,$$with$$H(x,p):=\underset{b\in B}{min}\underset{a\in A}{max}\{-f(x,a,b)·p-l(x,a,b)\}.$$We discuss the cases in which the state of the system is required to stay in an $n$-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain $\Omega \subset {ℝ}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary)...

We study the asymptotic behavior of $\lambda v_{\lambda }$ as $\lambda \to 0^{+}$, where $v_{\lambda }$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) $$\lambda v_{\lambda }+H(x,D{v}_{\lambda })=0,$$ with $$H(x,p):=\underset{b\in B}{min}\underset{a\in A}{max}\{-f(x,a,b)·p-l(x,a,b)\}.$$ We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain $\Omega \subset {}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the...

Noncooperative games with systems governed by nonlinear differential equations remain, in general, nonconvex even if continuously extended (i. e. relaxed) in terms of Young measures. However, if the individual payoff functionals are “enough” uniformly convex and the controlled system is only “slightly” nonlinear, then the relaxed game enjoys a globally convex structure, which guarantees existence of its Nash equilibria as well as existence of approximate Nash equilibria (in a suitable sense) for...

Two problems arising in Environment are considered. The first one concerns a conjecture posed by von Neumann in 1955 on the possible modification of the albedo in order to control the Earth surface temperature. The second one is related to the approximate controllability of Stackelberg-Nash strategies for some optimization problems as, for instance, the pollution control in a lake. The results of the second part were obtained in collaboration with Jacques-Lois Lions.

The aim of the present paper is to study problems of optimal design in mechanics, whose variational form is given by inequalities expressing the principle of virtual power in its inequality form. The elliptic, linear symmetric operators as well as convex sets of possible states depend on the control parameter. The existence theorem for the optimal control is applied to design problems for an elastic laminated plate whose variable thickness appears as a control variable.

We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear hamiltonian seems to be the...

We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be...