INFORMATION ABOUT PROJECT,
SUPPORTED BY RUSSIAN SCIENCE FOUNDATION

The information is prepared on the basis of data from the information-analytical system RSF, informative part is represented in the author's edition. All rights belong to the authors, the use or reprinting of materials is permitted only with the prior consent of the authors.

COMMON PART

Project Number24-21-00424

Project titleGeometric Properties of Attainability Sets in Solving Optimal Control Problems

Project LeadTarasyev Alexander

AffiliationN.N.Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences,

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-204 - Mathematical problems of control theory

Keywordsoptimal control, attainability set, Hamiltonian system, singular set, nonsmooth analysis, weakly convex set, alpha-set

PROJECT CONTENT

Annotation
The project deals with the theoretical foundations of effective numerical solution of control problems for dynamic systems based on the geometric properties of attainability sets. For example, if the attainability set of the control system at any time has a "good" geometry, i.e. is simply connected and has a sufficiently smooth boundary with small curvature, then it is sufficient to "track" only the change in the boundary of the attainability set, which actually reduces the dimension of the problem. Another example is the possibility of constructing optimal control switching surfaces based on the “weak” convexity properties of attainability sets or cost functions. The project considers a series of optimal control problems in which the property of non-convex attainability sets arises both as a result of nonlinear dynamics or the payoff functional, and in the case of non-convex target sets. It is supposed to study attainability sets and pencils of trajectories in control problems and differential games with an infinite horizon. Resolvable control procedures will be constructed based on nonlinear controllers for Hamiltonian systems arising in the maximum principle of L.S. Pontryagin. Approximation procedures for constructing attainability sets, cost functions, and optimal control strategies based on compactified grid schemes will be developed. It is planned to study time optimal control problems for dynamical systems in the three-dimensional space, which have simple dynamics and a non-convex target set with violation of the smoothness of the boundary. The resolving sets in such problems contain singularities, which are the areas of non-smoothness of the solution. The actual problems here are the problem of creation of new analytical methods for identifying singular sets from the point of view of the development of the theory, and the problem of development of new correct computational procedures for constructing approximations of solutions from the point of view of numerical algorithms. In addition to the degree of smoothness and the presence of singularities, the most important geometric characteristic of attainability sets is the degree of their nonconvexity, as well as the time during which the attainability set is guaranteed to retain the simply connected property. The degree of non-convexity in this project is proposed to be measured using the concept of an alpha set, which was formed in the early 2000s and is one of the generalizations of the concept of a convex set. This problem has already been considered by members of the research team of the project, the key lemma in solving this problem is the lemma on the numerical relationship between alpha sets and Vial weakly convex sets with constant R. However, the estimate obtained earlier depends on the dimension of the space (in the two-dimensional space, the accuracy of the estimate of the degree of nonconvexity is higher), so research team plan to either improve the key lemma for three-dimensional spaces or give examples proving that the estimates are unimprovable. For the practical application of the theory of alpha sets, it is important to have an algorithm for numerically calculating the degree of nonconvexity in terms of alpha sets. At present, such an algorithm has been developed by members of the scientific team only for polygons on a plane and without assessing its error. The research team will consider the problem of a proven estimate of the measure of non-convexity of a flat set specified "pixelwise" (i.e., in the form of a finite set of points, which is an approximation close to the original set in the Hausdorff metric) under the condition of simply connectedness and a known minimum radius of curvature of the boundary. The research team will also consider the control problems for specific dynamical systems, such as, for example, the "Dubins Car" and its analogues in higher-dimensional spaces. The construction of solutions in such problems will be carried out using resolving structures based on the attainability sets of the considered control systems.

Expected results
The project deals with the theoretical foundations of effective numerical solution of control problems for dynamic systems based on the geometric properties of attainability sets. For example, if the attainability set of the control system at any time has a "good" geometry, i.e. is simply connected and has a sufficiently smooth boundary with small curvature, then it is sufficient to "track" only the change in the boundary of the attainability set, which actually reduces the dimension of the problem. Another example is the possibility of constructing optimal control switching surfaces based on the “weak” convexity properties of attainability sets or cost functions. The project considers a series of optimal control problems in which the property of non-convex attainability sets arises both as a result of nonlinear dynamics or the payoff functional, and in the case of non-convex target sets. It is supposed to study attainability sets and pencils of trajectories in control problems and differential games with an infinite horizon. Resolvable control procedures will be constructed based on nonlinear controllers for Hamiltonian systems arising in the maximum principle of L.S. Pontryagin. Approximation procedures for constructing attainability sets, cost functions, and optimal control strategies based on compactified grid schemes will be developed. It is planned to study time optimal control problems for dynamical systems in the three-dimensional space, which have simple dynamics and a non-convex target set with violation of the smoothness of the boundary. The resolving sets in such problems contain singularities, which are the areas of non-smoothness of the solution. The actual problems here are the problem of creation of new analytical methods for identifying singular sets from the point of view of the development of the theory, and the problem of development of new correct computational procedures for constructing approximations of solutions from the point of view of numerical algorithms. In addition to the degree of smoothness and the presence of singularities, the most important geometric characteristic of attainability sets is the degree of their nonconvexity, as well as the time during which the attainability set is guaranteed to retain the simply connected property. The degree of non-convexity in this project is proposed to be measured using the concept of an alpha set, which was formed in the early 2000s and is one of the generalizations of the concept of a convex set. This problem has already been considered by members of the research team of the project, the key lemma in solving this problem is the lemma on the numerical relationship between alpha sets and Vial weakly convex sets with constant R. However, the estimate obtained earlier depends on the dimension of the space (in the two-dimensional space, the accuracy of the estimate of the degree of nonconvexity is higher), so research team plan to either improve the key lemma for three-dimensional spaces or give examples proving that the estimates are unimprovable. For the practical application of the theory of alpha sets, it is important to have an algorithm for numerically calculating the degree of nonconvexity in terms of alpha sets. At present, such an algorithm has been developed by members of the scientific team only for polygons on a plane and without assessing its error. The research team will consider the problem of a proven estimate of the measure of non-convexity of a flat set specified "pixelwise" (i.e., in the form of a finite set of points, which is an approximation close to the original set in the Hausdorff metric) under the condition of simply connectedness and a known minimum radius of curvature of the boundary. The research team will also consider the control problems for specific dynamical systems, such as, for example, the "Dubins Car" and its analogues in higher-dimensional spaces. The construction of solutions in such problems will be carried out using resolving structures based on the attainability sets of the considered control systems.

REPORTS