We consider the time optimal control problem for a semilinear parabolic control system, where the target is the closed ball with center 0 and radius $Rgeq0$ in a Hilbert space X. In particular, we ...allow the origin of X to be the target. Using an appropriate Kruzkov-type transformation, we give an existence and uniqueness result for the associated Hamilton--Jacobi--Bellman equation, even when the reachable set is not the whole space.
We consider here parabolic equations with infinitely many variables. By using only analytic tools we prove existence, uniqueness, and regularity of solutions.
In this paper we study a Hamilton-Jacobi equation related to the boundary control of a parabolic equation with Neumann boundary conditions. The state space of this problem is a Hilbert space and the ...equation is defined classically only on a dense subset of the state space. Moreover the Hamiltonian appearing in the equation contains fractional powers of an unbounded operator. These facts render the problem difficult. In this paper we give a revised definition of a viscosity solution to accommodate the unboundness of the Hamiltonian. We then obtain existence and uniqueness results for viscosity solutions. In particular we show that under suitable assumptions the value function of the boundary control problem is the unique viscosity solution of the related Hamilton-Jacobi equation.
In the dynamical theory of granular matter, the so-called table problem consists in studying the evolution of a heap of matter poured continuously onto a bounded domain Ω ⊂ R2. The mathematical ...description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of Ω.
Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically formula omitted, refer ...to PDF or formula omitted, refer to PDF. Considering an unbounded and disconnected control region of the form formula omitted, refer to PDF, we prove two null controllability results: under some technical assumption on the control parts formula omitted, refer to PDF, we prove that every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2(Ω) can be controlled to zero using control functions in a weighted L 2 space. At last we give several examples in which the control region has a finite measure and our null controllability results apply. PUBLICATION ABSTRACT
The optimal control of a distributed parameter system is connected to the solution of the corresponding Hamilton-Jacobi equation. This is a first-order equation in infinite dimensions with ...discontinuous coefficients. We study the Hamilton-Jacobi equation of a system governed either by a “semilinear” or by a “monotone” dynamics, replacing the unbounded terms of this equation by their Yosida approximations. We prove convergence of the approximate solutions to the value function of the original problem, by using the uniqueness result for viscosity solutions. Examples and applications are included.
Several characterizations of optimal trajectories for the classical Mayer problem in optimal control are provided. For this purpose the regularity of directional derivatives of the value function is ...studied: for instance, it is shown that for smooth control systems the value function $V$ is continuously differentiable along an optimal trajectory $x:t_0 ,1 \to {\bf R}^n $ provided $V$ is differentiable at the initial point $(t_0 ,x(t_0 ))$. Then the upper semicontinuity of the optimal feedback map is deduced. The problem of optimal design is addressed, obtaining sufficient conditions for optimality. Finally, it is shown that the optimal control problem may be reduced to a viability one.
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