Sabine Pickenhain

Second-order sufficient conditions for control problems with mixed control-state constraints

References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.

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.

On Adequate Transversality Conditions for Infinite Horizon Optimal Control Problems—A Famous Example of Halkin

In this paper we apply a duality concept of Klotzler (Equadiff IV. Proceedings of the Czechoslovak conference on differential equations and their applications held in Prague, 22–26 August, 1977, Lecture notes in mathematics, vol. 703, pp. 189–196, Springer, Berlin, 1979) to infinite horizon optimal control problems. The key idea is the choice of weighted Sobolev spaces as state spaces.

Pontryagin principle for state-constrained control problems governed by a first-order PDE system

This paper considers multidimensional control problems governed by a first-order PDE system and state constraints. After performing the standard Young measure relaxation, we are able to prove the Pontryagin principle by means of an ∈-maximum principle. Generalizing the common setting of one-dimensional control theory, we model piecewise-continuous weak derivatives as functions of the first Baire class and obtain regular measures as corresponding multipliers. In a number of corollaries, we derive necessary optimality conditions for local minimizers of the state-constrained problem as well as for global and local minimizers of the unconstrained problem.

Hilbert Space Treatment of Optimal Control Problems with Infinite Horizon

We consider a class of infinite horizon optimal control problems as optimization problems in Hilbert spaces. For typical applications it is demonstrated that the state and control variables belong to a Weighted Sobolev – and Lebesgue space, respectively. In this setting Pontryagin’s Maximum Principle as necessary condition for a strong local minimum is shown. The obtained maximum principle includes transversality conditions as well.

Pontryagin’s Maximum Principle for Multidimensional Control Problems

A weak maximum principle is shown for general problems n n

On a resource allocation model with infinite horizon

{text{minimize}},fleft( {x,{text{ }}w} right),,{text{on }}{X_0} times {X_{text{1}}},{text{with respect to}},linear,{text{state constraints}},{A_0}x = {A_{text{1}}}w

Budget-constrained infinite horizon optimal control problems with linear dynamics

n nin Banach spaces X 0 and local convex topological vector spaces X 1, where f(x, •) is a convex functional on X 1 and X j are linear and continuous operators from X j to a Hilbert space X (j = 0,1). The proved theorem is applied to Dieudonne-Rashevsky-type and relaxed control problems.