Maria do Rosário de Pinho

Mixed constrained control problems

Necessary optimality conditions are derived in the form of a weak maximum principle for optimal control problems with mixed state-control equality and inequality constraints. In contrast to previous work these conditions hold when the Jacobian of the active constraints, with respect to the unconstrained control variable, has full rank. A feature of these conditions is that they are stated in terms of a joint Clarke subdifferential. Furthermore the use of the joint subdifferential gives sufficiency for nonsmooth, normal, linear convex problems. The main point of interest is not only the full rank condition assumption but also the nature of the analysis employed in this paper. A key element is the removal of the constraints and application of Ekelands variational principle.

Weak maximum principle for optimal control problems with mixed constraints

Optimality conditions for control problems with mixed state-control constraints have been the focus of attention for a long time. In particular, the subject of necessary conditions in the form of maximum principles have been addressed by a number of authors; see for example [1–4], to name but a few. Weak maximum principles, which apply to weak local solutions, covering problems with possibly nonsmooth data, have been considered in [5] and, in a more general setting, in [3]. For nonsmooth problems, strong maximum principles, which in turn apply to “strong” local solutions, have also received some attention recently (see [6,7]). Various recent results, including those of the present paper, can be captured as special cases of the following optimal control problem with mixed constraints, also known as

Mixed constraints in optimal control: an implicit function theorem approach

According to a widely quoted paper by Gilbert and Bernstein (1983), ‘mathematical rigorous treatments of second order necessary conditions for problems in optimal control seem to be limited’. In that paper, two different sets of second-order conditions are obtained by applying necessary conditions for an abstract optimization problem developed by the second author (Bernstein, 1984). These conditions are compared with, and shown to be a generalization of, those previously derived by Hestenes (1966) and Warga (1978). In a brief summary of the problems considered in those two references, Gilbert and Bernstein write: ‘Hestenes, whose work is the earliest, considered a fairly general optimal control problem but made the standard assumption that the control set is open. His main result states that the second variation of a suitably defined function is nonnegative on a set of admissible variations related to the first order necessary conditions. More recently, Warga obtained a similar result, stated in a somewhat different way, for problems where the controls are restricted to a convex, not necessarily open, constrained set.’ Since then, this question has been studied by a large number of authors. In the literature, one can now find second-order conditions for a wide variety of different specific optimal control problems according to the constraints, the spaces of admissible processes, the assumptions on the functions delimiting the problem, and so on (see e.g. Arutyunov & Vereshchagina, 2002; Loewen & Zheng, 1994; Milyutin & Osmolovskii, 1998; Osmolovskii, 1975; Stefani & Zezza, 1996; Zeidan, 1994, 1996; Zeidan & Zezza, 1988, and references therein). Not all coincide, and it may be extremely cumbersome to compare between different problems and the conditions obtained, but this is not the issue of this paper.

An Optimal Control Algorithm For Multidrug Cancer Chemotherapy Design

In this paper, we present an apprmh to model the tumour growth and optimize a combination chemotherapy. We propose the use of in vitro data to estimate the parameter values of a multicompartmental model based on cell kinetics and incorporating the effect of drugs under consideration. The optimal control problem consists in finding the combination chemtherapy which minimizes a custom defined cost function, reflecting a compromise between toxicity effects and tumour growth, while satisfying dynamic and static constraints. A search direction algorithm seeking a multidrug dosage schedule satisfying the necessary conditions of optimality is constucted.

Necessary Conditions for Constrained Problems under Mangasarian-Fromowitz Conditions

The focus of this paper is on first order necessary conditions for optimal control problems with mixed state-control equality and inequality constraints. We consider the case when the cost and dynamics are nonsmooth, and the constraints satisfy Mangasarian-Fromowitz-type assumptions that weaken the commonly used hypothesis that the Jacobian of the active constraints, with respect to the free variable, is of full rank. The results are formulated as unmaximized Hamiltonian inclusion-type conditions involving not the customary product of partial subdifferentials but the joint subdifferential with respect to the state and control variables.

An Extension of the Schwarzkopf Multiplier Rule in Optimal Control

The search for multiplier rules in dynamic optimization has been an important theme in the subject for over a century; it was central in the classical calculus of variations, and the Pontryagin maximum principle of optimal control theory is part of this quest. A more recent thread has involved problems with so-called mixed constraints involving the control and state variables jointly, a subject which now boasts a considerable literature. Recently, Clarke and de Pinho proved a general multiplier rule for such problems that extends and subsumes rather directly most of the available results, namely, those which postulate some kind of rank condition or, more generally, a constraint qualification (or generalized Mangasarian-Fromowitz condition). An exception to this approach is due to Schwarzkopf, whose well-known theorem replaces the rank hypothesis, for relaxed problems, by one of covering. The purpose of this article is to show how to obtain this type of theorem from the general multiplier rule of Clarke and de Pinho. In so doing, we subsume, extend, and correct the currently available versions of Schwarzkopfs result.

On application of optimal control to SEIR normalized models: Pros and cons

In this work we normalize a SEIR model that incorporates exponential natural birth and death, as well as disease-caused death. We use optimal control to control by vaccination the spread of a generic infectious disease described by a normalized model with L1 cost. We discuss the pros and cons of SEIR normalized models when compared with classical models when optimal control with L1 costs are considered. Our discussion highlights the role of the cost. Additionally, we partially validate our numerical solutions for our optimal control problem with normalized models using the Maximum Principle.

Optimal control problems for path planing of AUV using simplified models

Here we propose a simplified model for the path planning of an Autonomous Underwater Vehicle (AUV) in an horizontal plane when ocean currents are considered. The model includes kinematic equations and a simple dynamic equation. Our problem of interest is a minimum time problem with state constraints where the control appears linearly. This problem is solved numerically using the direct method. We extract various tests from the Maximum Principle that are then used to validate the numerical solution. In contrast to many other literature we apply the Maximum Principle as defined in [9].

Optimal control for an irrigation planning problem: characterisation of solution and validation of the numerical results

In a previous study, the authors developed the planning of the water used in the irrigation systems of a given farmland in order to ensure that the field cultivation is in a good state of preservation. This planning was modelled and tackled as an optimal control problem: minimize the water flow (control) so that the extent water amount in the soil (trajectory) fulfils the cultivation water requirements. In this paper, we characterize the solution of our problem guaranteeing the existence of the solution and applying the necessary and sufficient conditions of optimality. We validate the numerical results obtained previously, comparing the analytical and numerical solutions.

An Optimal Control Framework for Resources Management in Agriculture

An optimal control framework to support the management and control of resources in a wide range of problems arising in agriculture is discussed. Lessons extracted from past research on the weed control problem and a survey of a vast body of pertinent literature led to the specification of key requirements to be met by a suitable optimization framework. The proposed layered control structure—including planning, coordination, and execution layers—relies on a set of nested optimization processes of which an “infinite horizon” Model Predictive Control scheme plays a key role in planning and coordination. Some challenges and recent results on the Pontryagin Maximum Principle for infinite horizon optimal control are also discussed.