arXiv subjects
Publications and source records attributed to Arkadiusz Misztela.
In this paper, for the Hamilton-Jacobi-Bellman equation with an infinite horizon and state constraints, we construct a suitably regular representation. This allows us to reduce the problem of existence and uniqueness of solutions to the Frankowska and Basco theorem from (2019). Furthermore, we demonstrate that our representations are stable. The obtained results are illustrated with examples.
This article is devoted to the study of lower semicontinuous solutions of Hamilton-Jacobi equations with convex Hamiltonians in a gradient variable. Such Hamiltonians appear in the optimal control theory. We present a necessary and sufficient condition for a reduction of a Hamiltonian satisfying optimality conditions to the case when the Hamiltonian is positively homogeneous and also satisfies optimality conditions. It allows us to reduce some uniqueness problems of lower semicontinuous solutions to Barron-Jensen and Frankowska theorems. For Hamiltonians, which cannot be reduced in that way, we prove the new existence and uniqueness theorems.
An example of a nonunique solution of the Cauchy problem of Hamilton-Jacobi-Bellman (HJB) equation with surprisingly regular Hamiltonian is presented. The Hamiltonian H(t,x,p) is locally Lipschitz continuous with respect to all variables, convex in p and with linear growth with respect to p and x. The HJB equation possesses two distinct lower semicontinuous solutions with the same final conditions; moreover, one of them is the value function of the corresponding Bolza problem. The definition of lower semicontinuous solution was proposed by Barron-Jensen (1990) and Frankowska (1993). Using the example an analysis and comparison of assumptions in some uniqueness results in HJB equations is provided.
In this paper we study the existence of sufficiently regular representations of Hamilton-Jacobi equations in the optimal control theory with unbounded control set. We use a new method to construct representations for a wide class of Hamiltonians. This class is wider than any constructed before, because we do not require Legendre-Fenchel conjugates of Hamiltonians to be bounded. However, in this case we obtain representations with unbounded control set. We apply the obtained results to study regularities of value functions and correlations between variational and optimal control problems.
We describe the family of initial conditions for Hamilton-Jacobi equations (HJE) corresponding to optimal control problems that can be retrieved by solving HJE backward in time.
In this paper we study the existence of sufficiently regular representations of Hamilton-Jacobi equations in optimal control theory with the compact control set. We introduce a new method to construct representations for a wide class of Hamiltonians, wider than it was achieved before. Our result is proved by means of these conditions on Hamiltonian that are necessary for the existence of a representation. In particular, we solve an open problem of Rampazzo (2005). We apply the obtained results to reduce a variational problem to an optimal control problem.
In the paper we study the following problem: given a Hamilton-Jacobi equation where the Hamiltonian is convex with respect to the last variable, are there any optimal control problems representing it? In other words, we search for an appropriately regular dynamics and a Lagrangian that represents the Hamiltonian with given properties. This problem was lately researched by Frankowska-Sedrakyan (2014) and Rampazzo (2005). We introduce a new method to construct a representation of a wide class of Hamiltonians, wider than it was achieved before. Actually, we get two types of representations: with compact and noncompact control set, depending on regularity of the Hamiltonian. We conclude the paper by proving the stability of representations.