Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost

We study viscosity solutions of Hamilton-Jacobi equations that arise in optimal control problems with unbounded controls and discontinuous Lagrangian. In our assumptions, the comparison principle will not hold, in general. We prove optimality principles that extend the scope of the results of [23] under very general assumptions, allowing unbounded controls. In particular, our results apply to calculus of variations problems under Tonelli type coercivity conditions. Optimality principles can be applied to obtain necessary and sufficient conditions for uniqueness in boundary value problems, and to characterize minimal and maximal solutions when uniqueness fails. We give examples of applications of our results in this direction.     

On unbounded solutions of Bellman's equation associated with optimal switching control problems with state constraints

We study a quasi-variational inequality system with unbounded solutions. It represents the Bellman equation associated with an optimal switching control problem with state constraints arising from production engineering. We show that the optimal cost is the unique viscosity solution of the system.This work was supported by the National Research Council of Argentina, Grant No. PID-BID 213.     

On some two-point problems for a fourth-order equation

We obtain sufficient conditions for the existence of a solution of two boundary value problems for a nonlinear fourth-order equation of general form.     

On Bellman function for extremal problems in BMO

In this Note we describe our results on construction of the Bellman function solving an extremal problem for a large class of integral functionals on BMO.     

The Bellman equation for time-optimal control of noncontrollable,nonlinear systems

For a general nonlinear system and closed target set we study the value functions and of the control problems of reaching and, respectively, its interior, in minimum time. Under no controllability assumptions on the system, we characterize them as, respectively, the minimal viscosity supersolution and the maximal viscosity subsolution of the Bellman equation with appropriate boundary conditions. Then we prove that is the unique upper semicontinuous complete solution of such a boundary value problem, which means in particular that the (completed) graph of contains the graph of any solution, as well as all the limits of reasonable approximating sequences. We give some applications to verifications theorems and to the stability of the minimum time function with respect to general perturbations.The authors are partially supported by the Italian National Projects Equazioni di evoluzione e applicazioni fisico-matematiche and Equazioni differenziali e calcolo delle variazioni, respectively.     

On the Bellman equation of the overtaking criterion: Addendum

In a previous note, we discussed the properties of solutions to the Bellman equation of the Gale overtaking criterion. The purpose of this note is to show that the dynamic programming approach may also be used for the Brock criterion.We are indebted to an anonymous referee who contributed to improve this paper.     

On the lindblad equation with unbounded time-dependent coefficients

We prove newa priori estimates for the resolvent of a minimal quantum dynamical semigroup. These estimates simplify well-known conditions sufficient for conservativity and impose continuity conditions on the time-dependent operator coefficients ensuring the existence of conservative solutions of the Markov evolution equations. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 125–140, January, 1997. Translated by A. M. Chebotarev     

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton–Jacobi–Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear–quadratic model.     

On the terminal condition for the Bellman equation for dynamic optimization with an infinite horizon

In this work a sufficient condition for deterministic dynamic optimization with discrete time and infinite horizon is formulated. It encompasses also situations where the instantaneous payoff/utility function can attain infinite values.The usual terminal condition for sufficiency of the Bellman equation requiring that the limit superior of the value function along each admissible trajectory is equal to 0 is replaced by a weaker one in which the limit superior of the value function can attain nonpositive values.This kind of terminal condition is applicable also to deterministic dynamic optimization problems with real-valued instantaneous payoff function in which the usual terminal condition does not hold.     

Optimal control problems with unbounded constraint sets

We consider in this paper optimal control problems in which some of the constraint sets are unbounded. Firstly we deal with problems in which the control set is unbounded, so that ‘impulses’ are allowed as admissible controls, discontinuous functions as admissible trajectories. The second type of problem treated is that of infinite horizons, the time set being unbounded. Both class of problems are treated in a similar way. Firstly, a problem is transformed into a semi-infinite linear programming problem by embedding the spacesof admissible trajectory-control pairs into spaces of measures. Then this is mapped into an appropriate nonstandard structure, where a near-minimizer is found for the non-standard optimization; this entity is mapped back as a minimizer for the original problem. An appendix is including introducing the basic concepts of nonstandard analysis

Numerical methods are presented for the estimation of the minimizing measure, and the construction of nearly optimal trajectory-control pairs. Examples are given involving multiplicative controls     

On some classes of unbounded operators

Hyponormality, normality and subnormality for unbounded operators on Hilbert space are investigated and quasi- similarity of such operators is discussed.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Existence theorems for lagrange control problems with unbounded time domain

Existence theorems are proved for usual Lagrange control systems, in which the time domain is unbounded. As usual in Lagrange problems, the cost functional is an improper integral, the state equation is a system of ordinary differential equations, with assigned boundary conditions, and constraints may be imposed on the values of the state and control variables. It is shown that the boundary conditions at infinity require a particular analysis. Problems of this form can be found in econometrics (e.g., infinite-horizon economic models) and operations research (e.g., search problems).The author wishes to thank Professor L. Cesari for his many helpful comments and assistance in the preparation of this paper. This work was sponsored by the United States Air Force under Grants Nos. AF-AFOSR-69-1767-A and AFOSR-69-1662.