58 research outputs found
Existence of optimal nonanticipating controls in piecewise deterministic control problems
Optimal nonanticipating controls are shown to exist in nonautonomous piecewise deterministic control problems with hard terminal restrictions. The assumptions needed are completely analogous to those needed to obtain optimal controls in deterministic control problems. The proof is based on well-known results on existence of deterministic optimal controls
Piecewise deterministic optimal control problems
Piecewise deterministic control problems are problems involving stochastic disturbance of a special type. In certain situations, in an otherwise deterministic control system, it may happen that the state jumps at certain stochastic points of time. Examples are sudden oil finds, or sudden discoveries of metal deposits. Similarly, in seemingly deterministic processes, the dynamics may suddenly change character: at certain stochastic points in time, the right-hand side of the differential equation governing the system changes form, such changes being effected by jumps in a (dummy) state variable. Examples of such phenomena are sudden inventions, sudden ecological disasters, earthquakes, floods, storms, fires, the sudden capture of a criminal, that suddenly change the prospects of the firm, the society, the agriculture, the criminal... Several papers have discussed such problems, often using more or less ad hoc methods. (Sometimes it is possible to rewrite the problem so that deterministic control theory applies). A systematic method for solving such problems, based on HJB-equation (the Hamilton-Jacoby-Bellman equation) for the problem, is presented in Davis (1993). Markov Models and Optimization, and also briefly discussed below. In this paper a related method, closer to deterministic control theory, is presented first. It is easiest to apply to problems with a bound on the number of possible jumps. Thus, the main purpose of this paper is to show how some piecewise deterministic optimal control problems can be solved by techniques similar to those used in deterministic problems. The paper includes statements of several theoretical results. Proofs are given for the results involving the HJB-equation and fields of extremals, (for the HJB-equation, replicating the ones in Davis (1993))
Pareto improvements of Nash equilibria in differential games
This paper yields sufficient conditions for Pareto inoptimality of controls forming Nash equilibria in differential games. In Appendix a result on existence of open loop Nash equilibria is added
Open mapping theorems for directionally differentiable functions
Open mapping theorems are proved for directionally differentiable Lipschitz continuous functions. It is indicated that generalizations to nonsmooth functions that are not directionally differentiable are possible. The results in the paper generalize the open mapping theorems for differentiable mappings, and are different from open mapping theorems for nonsmooth functions in the literature, when these are specialized to directionally differentiable functions
Conditions implying the vanishing of the Hamiltonian at the infinite horizon in optimal control problems
n an infinite horizon optimal control problem, the Hamiltonian vanishes at the infinite horizon, when the differential equation is autonomous. The integrand in the integral criterion may contain the time explicitly, but it has to satisfy certain integrability conditions. A generalization of Michels (1982) result is obtained
Maximum principle for stochastic control in continuous time with hard end constraints
A maximum principle is proved for certain problems of continuous time stochastic control with hard end constraints, (end constraints satis_ed a.s.) After establishing a general theorem, the results are applied to problems where the state equation (di_erential equation) changes at certain stochastic points in time, and to piecewise continuous stochastic problems (including piecewise deterministic problems)
Discontinuous control systems
By means of some simple examples from economics, we elucidatecertain solution tools for the solution of optimal control problems were the system under study undergoes major changes when certain boundaries are crossed. The major changes may be that the state gets a jump discontinuity when crossing a boundary, or that the right hand side of the differential equation changes. Some theoretical result are presented. Among the results presented, at least the sufficient condition related to fields of extremals should be new
Existence of optimal nonanticipating controls in piecewise deterministic control problems
Optimal nonanticipating controls are shown to exist in nonautonomous piecewise
deterministic control problems with hard terminal restrictions. The assumptions needed are
completely analogous to those needed to obtain optimal controls in deterministic control
problems. The proof is based on well-known results on existence of deterministic optimal
controls
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