M.d.R. de Pinho

An Euler–Lagrange Inclusion for Optimal Control Problems with State Constraints

New first-order necessary conditions for optimality for control problems with pathwise state constraints are given. These conditions are a variant of a nonsmooth maximum principle which includes a joint subdifferential of the Hamiltonian – a condition called Euler–Lagrange inclusion (ELI). The main novelty of the result provided here is the ability to address state constraints while using an ELI.The ELI conditions have a number of desirable properties. Namely, they are, in some cases, able to convey more information about minimizers, and for the normal convex problems they are sufficient conditions of optimality. It is shown that these strengths are retained in the presence of state constraints.

An integral-type constraint qualification to guarantee nondegeneracy of the maximum principle for optimal control problems with state constraints

Abstract For optimal control problems involving ordinary differential equations and functional inequality state constraints, the maximum principle may degenerate , producing no useful information about minimizers. This is known as the degeneracy phenomenon. Several non-degenerate forms of the maximum principle, valid under different constraint qualifications, have been proposed in the literature. In this paper we propose a new constraint qualification under which a nondegenerate maximum principle is validated. In contrast with existing results, our constraint qualification is of an integral type. An advantage of the proposed constraint qualification is that it is verified on a larger class of problems with nonsmooth data and convex velocity sets.

A variant of nonsmooth maximum principle for state constrained problems

We derive a variant of the nonsmooth maximum principle for problems with pure state constraints. The interest of our result resides on the nonsmoothness itself since, when applied to smooth problems, it coincides with known results. Remarkably, in the normal form, our result has the special feature of being a sufficient optimality condition for linear-convex problems, a feature that the classical Pontryagin maximum principle had whereas the nonsmooth version had not. This work is distinct to previous work in the literature since, for state constrained problems, we add the Weierstrass conditions to adjoint inclusions using the joint subdifferentials with respect to the state and the control. Our proofs use old techniques developed in [16], while appealing to new results in [7].

Optimal control problems with nonsmooth mixed constraints

In this paper we present a weak maximum principle for optimal control problems involving mixed constraints and pointwise set control constraints. Notably such result holds for problems with possibly nonsmooth mixed constraints. Although the setback of such result resides on a convexity assumption on the ¿extended velocity set¿, we show that if the number of mixed constraints is one, such convexity assumption may be removed when an interiority assumption holds.

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

ABSTRACT This paper presents a new approach for the efficient solution of singular optimal control problems (SOCPs). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution. At first, the SOCP is converted into a binary optimal control problem. Then, by utilising the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the optimal control and to compute the approximating solution. The main advantages of the present method are that: (1) without a priori information, the structure of optimal control is detected; (2) it produces good results even using a small number of collocation points; (3) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.

Free time and mixed constrained optimal control problems

We consider free time optimal control problems with pointwise set control constraints u(t) ∈ U(t). Here we derive necessary conditions of optimality for those problem where the set U(t) is defined by equality and inequality control constraints. The main ingredients of our analysis are a well known time transformation and recent results on necessary conditions for mixed state-control constraints.

On Optimality Conditions for Control Problems with Constraints

Abstract We report on optimality conditions for control problems with mixed state control constraints and pure state constraints. Our main goal is to unify previously developed work and to illustrate the different approaches used. We describe necessary conditions that include an Euler-Lagrange inclusion for a weak minimizer as well as necessary conditions in the form of a maximum principle for a strong minimizer. The sufficiency of the same conditions for a certain class of these problems is also analysed

A maximum principle for optimal control problems with equality and inequality constraints

Optimal control problems involving state-dependent control constraints in the form of equalities and inequalities are considered. Necessary conditions of optimality are derived in the form of a weak maximum principle. It applies to problems with nonsmooth data.Optimal control problems involving state-dependent control constraints in the form of equalities and inequalities are considered. Necessary conditions of optimality are derived in the form of a weak maximum principle. It applies to problems with nonsmooth data.

Optimal control problems with state-dependent control constraints

We consider optimal control problems with state-dependent control constraints to which we associate a differential inclusion control problems. Necessary optimality conditions are obtained for the latter using proximal normal analysis. It is then shown that calmness guarantees the nontriviality of such conditions. The calmness assumption on the differential inclusion problem permits the derivation of optimality conditions in the form of a maximum principle for the original problem.<<ETX>>