Special Session 68: Optimal control theory and applications

Necessary conditions of optimality for dynamic optimization problems involving parameters

Piernicola Bettiol University of Brest France Co-Author(s):

Abstract: Optimal control problems with parameters provide a natural modelling framework for a wide range of applications, including aerospace engineering, machine learning, and biological systems. Depending on the underlying model and objectives, different performance criteria may be adopted. For instance, in some circumstances, average-cost formulations offer an alternative to classical minimax or robust optimization. Such problems may exhibit even multi-level structures, for example when an optimal control problem is coupled with a nonlinear programming problem arising from parameter optimization or estimation. In this talk, we introduce classes of parameter-dependent optimal control problems in which unknown parameters appear in the system dynamics, the cost function, endpoint constraints, and possibly even in the state constraints. For the different cost criteria under consideration, we derive and discuss the corresponding necessary conditions of optimality.

Set-valued Lie brackets and non-smooth controllability

Rampazzo Franco University of Padova Italy Co-Author(s):    Franco Rampazzo, Ermal Feleqi, Rohit Gupta

Abstract: {\it Small time local controllability} for a control system of ODEs consists in the fact that, for every sufficiently small $t>0$, the points reached at a time $s\leq t$ by the trajectories of the system starting from a point $x_*$ form a full neighborhood of $x_*$. In the case of a dynamics which is (nonlinear but) linear in the controls, the celebrated Rashevskii-Chow`s theorem states that the so-called {\it full rank-condition} --namely the fact that iterated Lie brackets of the involved vector fields generate the whole tangent space-- is sufficient for small time local controllability. Rashevskii-Chow`s theorem is classically obtained under hypotheses of $C^\infty$ regularity. In this talk I will illustrate how this result (and other similar ones) can be extended by means of a notion of {\it set-valued Lie bracket} up to the point that tha highest order brackets involved in the full rank condition are just almost everywhere defined, bounded, maps. There are several applications of such controllability results, e.g. in the study of subelliptic pde`s and of Carnot-Carath\`eodory spaces. In particular, the presented result might represent a basis for a highly non-smooth subRiemannian geometry.

A Two-Stage Optimal Control Problem in the Lotka-Volterra Competition Model for Cancer Treatment

Ellina Grigorieva Texas Woman`s University USA Co-Author(s):    Evgenii Khailov

Abstract: Traditional continuous hormonal therapy for metastatic prostate cancer is often ineffective because it unintentionally favors drug-resistant cells by eliminating their drug-sensitive competitors. To address this issue, intermittent therapy, which involves cyclic periods of drug dosing and withdrawal, is used. This approach preserves the population of drug-sensitive cells, which, by competing for shared resources, suppress the proliferation of resistant strains. In this study, we formulate and analyze an optimal control problem using the Lotka-Volterra competition model. The model includes a control function for regulating transitions between therapeutic stages to determine an optimal solution that reproduces the effectiveness of intermittent regimens in balancing cancer cell dynamics. Optimal prostate cancer treatment protocols are derived and their application in real-world clinical practice is discussed.

Young differential inclusions and their properties

Mariusz Michta Institute of Mathematics, University of Zielona Gora Poland Co-Author(s):    Mariusz Michta

Abstract: In the talk, we present new types of differential inclusions with a set-valued Young integral, which generalize a single-valued Young differential equation. In the single-valued case, the Young integral has been used in a wide range of applications. In particular, one can consider stochastic equations with respect to non-semimartingale integrators, such as the Mandelbrot fractional Brownian motion, which has H\{o}lder continuous sample paths. Thus, it seems reasonable to investigate differential inclusions driven by such a new type of integral. In the presentation, we shall establish the main properties of solution sets of Young differential inclusions. In particular, solutions to an abstract optimization problem related to the inclusion will be presented. References: 1. M. Michta, J. Motyl, Selection properties and set-valued Young integrals of set-valued functions, Results Math. 75, 164 (2020). 2. M. Michta, J. Motyl, Set-valued functions of bounded generalized variation and set-valued Young integrals, J. Theor. Probab. 35 (2022), 528-549. 3. M. Michta, J. Motyl, Solution sets for Young differential inclusions, Qual. Theory Dyn. Syst. 22, 132 (2023). 4. M. Michta, J. Motyl, Properties of set-valued Young integrals and Young differential inclusions generated by sets of H\{o}lder functions, Nonliner Differ. Equ. Appl. 31, article no 70 (2024).

Minimizers that are not impulsive minimizers and higher order abnormality

Monica Motta Department of Mathematics, University of Padova Italy Co-Author(s):    Michele Palladino and Franco Rampazzo

Abstract: This talk addresses two related problems in optimal control. The first investigation consists of compatibility issues between two classical approaches to deriving necessary conditions for optimal control problems with a final target: the set-separation approach and penalization techniques. These methods generally lead to non-equivalent conditions, mainly due to their reliance on different notions of tangency at the target. We address this issue by considering Quasi Differential Quotient (QDQ) approximating cones (which are fit for the set-separation approach) and identifying conditions under which the Clarke tangent cone (which is a typical tool within penalization techniques) is also a QDQ approximating cone. In particular, we show that this property holds under suitable local invariance assumptions or when the target coincides locally with an r-prox regular set. Then, we apply this compatibility result to the study of infimum-gap phenomena in optimal control problems with unbounded controls and impulsive extensions. In particular, we establish a connection between the occurrence of infimum gaps for strict-sense minimizers and abnormality in a higher-order Maximum Principle involving Lie brackets. While the abnormality-gap correspondence beyond first-order conditions has been already established for impulsive minimizers, the utilization of the above compatibility issues allow us to extend this correspondence to strict-sense minimizers.

Self-Dual Approximations in Fully Convex Control Problems

Peter Wolenski Louisiana State University USA Co-Author(s):

Abstract: We present an approximation technique to Fully Convex Bolza problems using Goebel`s self-dualizing envelope that preserves the duality structure.