Valeriya Lykina

Sufficiency Conditions for Infinite Horizon Optimal Control Problems

In this paper we formulate and use the duality concept of Klotzler (1977) for infinite horizon optimal control problems. The main idea is choosing weighted Sobolev and weighted Lp spaces as the state and control spaces, respectively. Different criteria of optimality are known for specific problems, e.g. the overtaking criterion of von Weizsacker (1965), the catching up criterion of Gale (1967) and the sporadically catching up criterion of Halkin (1974). Corresponding to these criteria we develop the duality theory and prove sufficient conditions for local optimality. Here we use some remarkable properties of weighted spaces. An example is presented where the solution is obtained in the framework of these weighted spaces, but which does not belong to standard Sobolev spaces.

An Existence Theorem for a Class of Infinite Horizon Optimal Control Problems

In this paper, we deal with infinite horizon optimal control problems involving affine-linear dynamics and prove the existence of optimal solutions. The innovation of this paper lies in the special setting of the problem, precisely in the choice of weighted Sobolev and weighted Lebesgue spaces as the state and control spaces, respectively, which turns out to be meaningful for various problems. We apply the generalized Weierstraß theorem to prove the existence result. A lower semicontinuity theorem which is needed for that is shown under weakened assumptions.

An indirect pseudospectral method for the solution of linear-quadratic optimal control problems with infinite horizon

We consider a class of linear-quadratic infinite horizon optimal control problems in Lagrange form involving the Lebesgue integral in the objective. The key idea is to introduce weighted Sobolev spaces as state spaces and weighted Lebesgue spaces as control spaces into the problem setting. Then, the problem becomes an optimization problem in Hilbert spaces. We use the weight functions in our consideration. This problem setting gives us the possibility to extend the admissible set and simultaneously to be sure that the adjoint variable belongs to a Hilbert space too. For the class of problems proposed, existence results as well as a Pontryagin-type Maximum Principle, as necessary and sufficient optimality condition, can be shown. Based on this principle we develop a Galerkin method, coupled with the Gauss–Laguerre quadrature formulas as discretization scheme, to solve the problem numerically. Results are presented for the introduced model and different weight functions.

On a resource allocation model with infinite horizon

In this paper we treat a resource allocation model defined on an infinite interval. We show that the solution of the corresponding problem with finite horizon cannot be extended to a solution of the infinite horizon problem, since the resource allocation problem in the unmodified setting does not have a solution on an unbounded interval. To change this situation we bring an additional state constraint into the model which contains a weight function. The new problem, called now the adapted resource allocation problem, has an optimal solution which will be identified by means of the duality concept of Klotzler.

Budget-constrained infinite horizon optimal control problems with linear dynamics

In this paper a class of infinite horizon optimal control problems with an isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamic is allowed. The problem setting includes a weighted Sobolev space as the state space. We investigate the question of existence of an optimal solution and formulate a corresponding theorem. Which influence the isoperimetrical constraint may have on the feasible set and on the existence of an optimal solution is illustrated in details at hand of a linear-quadratic regulator model.

Infinite horizon cancer treatment model with isoperimetrical constraint: existence of optimal solutions and numerical analysis

ABSTRACT In this paper, a class of infinite horizon optimal control problems with a mixed control-state isoperimetrical constraint, also interpreted as a budget constraint, is considered. The underlying dynamics is assumed to be affine-linear in control. The crucial idea which is followed in this paper is the choice of a weighted Sobolev space as the state space. For this class of problems, we establish an existence result and apply it to a bilinear model of optimal cancer treatment with an isoperimetrical constraint including the overall amount of drugs used during the whole therapy horizon. A numerical analysis of this model is provided by means of open source software package OCMat, which implements a continuation method for solving discounted infinite horizon optimal control problems.

Existence Theorem for Infinite Horizon Optimal Control Problems with Mixed Control-State Isoperimetrical Constraint

In this paper a class of infinite horizon optimal control problems with a mixed control-state isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamics is allowed. The problem setting includes a weighted Sobolev space as the state space. For this class of problems, we establish an existence theorem. The proved theoretical result is applied to a mixed control-state budget constrained advertisement model.