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1
Content available Pontryagin’s maximum principle for optimal control problems governed by nonlinear impulsive differential equations
Leiva Hugo
Journal of Mathematics and Applications
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2023
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Vol. 46
15-68
EN
In this paper, we derive the Pontryagin’s maximum principle for optimal control problems governed by nonlinear impulsive differential equations. Our method is based on Dubovitskii-Milyutin theory, but in doing so, we assumed that the linear variational impulsive differential equation around the optimal solution is exactly controllable, which can be satisfied in many cases. Then, we consider an example as an application of the main result. After that, we study the case when the differential equation is of neutral type. Finally, several possible problems are proposed for future research where the differential equation, the constraints, the time scale, the impulses, etc. are changed.
2
Content available Network optimality conditions
Osmolovskii Nikolai P., Qian Meizhi, Sokołowski Jan
Control and Cybernetics
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2023
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Vol. 52, No. 2
129--180
EN
Optimality conditions for optimal control problems arising in network modeling are discussed. We confine ourselves to the steady state network models. Therefore, we consider only control systems described by ordinary differential equations. First, we derive optimality conditions for the nonlinear problem for a single beam. These conditions are formulated in terms of the local Pontryagin maximum principle and the matrix Riccati equation. Then, the optimality conditions for the control problem for networks posed on an arbitrary planar graph are discussed. This problem has a set of independent variables xi varying within their intervals [0, li], associated with the corresponding beams at network edges. The lengths li of intervals are not specified and must be determined. So, the optimization problem is non-standard, it is a combination of control and design of networks. However, using a linear change of the independent variables, it can be reduced to a standard one, and we show this. Two simple numerical examples for the single-beam problem are considered.
3
Content available Shape sensitivity of optimal control for the Stokes problem
Abdelbari Merwan, Nachi Khadra, Sokolowski Jan, Szulc Katarzyna
Control and Cybernetics
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2020
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Vol. 49, No. 1
11--40
EN
In this article, we study the shape sensitivity of optimal control for the steady Stokes problem. The main goal is to obtain a robust representation for the derivatives of optimal solution with respect to smooth deformation of the flow domain. We introduce in this paper a rigorous proof of existence of the material derivative in the sense of Piola, as well as the shape derivative for the solution of the optimality system. We apply these results to derive the formulae for the shape gradient of the cost functional; under some regularity conditions the shape gradient is given according to the structure theorem by a function supported on the moving boundary, then the numerical methods for shape optimization can be applied in order to solve the associated optimization problems.
4
Content available Existence and uniqueness of the solution to the optimal control problem with integral criterion over the entire domain for a nonlinear Schrödinger equation with a special gradient term
Salmanov Vuqar
Control and Cybernetics
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2020
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Vol. 49, No. 3
277--290
EN
The paper concerns the optimal control problem with the full-range integral performance criterion for the nonlinear Schrödinger equation with the specific gradient summand and the complex potential when the performance criterion is the full-range integral. In this paper, the existence and uniqueness theorems regarding the solution of the optimal control problem under consideration are proven.
5
Content available On a finding the coefficient of one nonlinear wave equation in the mixed problem
Safarova Zimrud R.
Archives of Control Sciences
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2020
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Vol. 30, No. 2
199--212
EN
The paper is devoted to the finding of the coefficient of one nonlinear wave equation in the mixed problem. The considered problem is reduced to the optimal control problem with proper functional. Differentiability of functional is proved and the necessary optimality conditions are derived in the form of the variational inequality. Existence of the optimal control is proved.
6
Content available The structure of the optimal control in the problems of strength optimization of steel girders
Mikulski Leszek
Archives of Civil Engineering
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2019
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Vol. 65, nr 4
277--293
EN
The paper concerns a strength optimization of continuous beams with variable cross-section. The continuous beams are subjected to a dead weight and a useful load, the six (seven) combinations of loads were analyzed. Optimal design problems in structural mechanics can by mathematically formulated as optimal control tasks. To solve the above formulated optimization problems, the minimum principle was applied. The paper is an introductory and survey paper of the treatment of realistically modelled optimal control problems from application in the structural mechanics. Especially those problems are considered, which include different types of constraints. The optimization problem is reduced to the solution of multipoint boundary value problems (MPBVP) composed of differential equations. Dimension of MPBVP is usually a large number, what produces numerical difficulties. Optimal control theory does not give much information about the control structure. The correctness of the assumed control structure can be checked after obtaining the solution of the boundary problem.
PL
Praca dotyczy optymalizacji wytrzymałościowej dźwigarów ciągłych trój-, cztero- i pięcioprzęsłowych o zmiennym dwuteowym przekroju poprzecznym. Dźwigary obciążone są ciężarem własnym i kombinacją obciążeń użytkowych (sześć lub siedem kombinacji). Deformacja dźwigara opisana jest przez układ równań różniczkowych z warunkami początkowymi i brzegowymi, ponadto do spełnienia pozostają wewnętrzne warunki brzegowe i warunki nieciągłości w pośrednich punktach podparcia. Rozważane są ograniczenia geometryczne, ograniczenia naprężeń i przemieszczeń. Jako funkcję celu wybrano objętość stali. Problemy optymalnego kształtowania formułowane są jako zadania teorii sterowania. Do rozwiązania zadań zaproponowano zasadę minimum. Problem optymalizacji redukuje się do rozwiązania wielopunktowego problemu brzegowego (WPPB) dla układu równań różniczkowych. Wymiar WPPB jest zwykle duży, co wymaga pokonania trudności numerycznych. Teoria sterowania nie dostarcza bowiem informacji o strukturze optymalnego rozwiązania dla której problem jest zbieżny. W pracy struktura sterowania opisuje kolejność występowania przedziałów i punktów z aktywnymi ograniczeniami. Poprawne przyjęcie tej struktury w rozwiązanych problemach jest zasadniczym osiągnięciem pracy. Uzyskane i prezentowane na wykresach wybrane zmienne stanu, zmienne sprzężone, zmienne decyzyjne, funkcje Hamiltona potwierdzają spełnienie warunków koniecznych optymalizacji.
7
Content available A Nash equilibrium approach for multiobjective optimal control problems with elliptic partial differential equations
Dreves A.
Control and Cybernetics
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2016
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Vol. 45, no. 4
457--482
EN
We consider the generalized Nash equilibrium as a solution concept for multiobjective optimal control problems governed by elliptic partial differential equations with constraints not only for the control but also for the state variables. In the first part, we present a constructive proof of the existence of a generalized Nash equilibrium via an approximating sequence of suitable finite dimensional discretizations. In the second part, we propose a variant of a potential reduction algorithm for the numerical solution of these discretized problems. In contrast to the existing numerical approaches ours does not require the computation of the control–to–state mapping. Instead we introduce different state variables and guarantee that they become equal at a solution. We prove sufficient conditions for the convergence of our algorithm to a solution. Furthermore, some numerical results showing the applicability are provided.
8
Content available remote Duality on geodesics of Cartan distributions and sub-Riemannian pseudo-product structures
Ishikawa G., Kitagawa Y., Yukuno W.
Demonstratio Mathematica
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2015
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Vol. 48, nr 2
193--216
EN
Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown in [15], that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.
9
Content available Optimal control of a fractional-order enzyme kinetic model
Al-Basir F., Elaiw A. M., Kesh D., Roy P. K.
Control and Cybernetics
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2015
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Vol. 44, no. 4
443--461
EN
Enzymes play a significant role in controlling the characteristics of various chemical and biochemical reactions. They act as catalysts that increase the rate of reaction without undergoing any change in quantity. Enzymatic reactions occur through the active sites, which combine with the substrates to form intermediate complexes, subsequently leading to products. An enzyme having two active sites can show cooperative phenomena. Against this background, an enzyme-kinetic mathematical model is formulated using fractional order derivatives. Optimal control mechanism has been incorporated into the fractional-order model system to maximize the product output. Euler-Lagrange optimality conditions are derived for the FOCP (fractional order control problem) using maximum principle. Numerical iterative schemes have been developed to solve the fractional order optimal control problem through Matlab.
10
Content available Continuous dependence of solutions of elliptic BVPs on parameters
Orpel A.
Opuscula Mathematica
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2006
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Vol. 26, no. 2
351-359
EN
The continuous dependence of solutions for a certain class of elliptic PDE on functional parameters is studied in this paper. The main result is as follow: the sequence {k}k∈N of solutions of the Dirichlet problem discussed here (corresponding to parameters {uk}k∈N) converges weakly to x0 (corresponding to u0) in W1,q 0 (Ω, R), provided that {uk}k∈N tends to u0 a.e. in Ω. Our investigation covers both sub and superlinear cases. We apply this result to some optimal control problems.
11
Content available Dual sufficient optimality conditions for optimal control
Młynarska E.
Control and Cybernetics
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2001
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Vol. 30, no 1
45-54
EN
In this paper we develop the first and second order dual sufficient optimality conditions for a nonlinear optimal control problem. Our conditions are derived from the dual Hamilton-Jacobi approach applied to the generalized problem of Bolza. We do not require neither any convexity on the data, nor that the control set U be polyhedral, nor that the control function be in the interior of U. Instead, we assume the existence of a function which satisfies a certain inequality.
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