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1
Content available Extremal Problems for Second Order Hyperbolic Systems with Boundary Conditions Involving Multiple Time-Varying Delays
Kowalewski Adam
Pomiary Automatyka Robotyka
|
2023
|
R. 27, nr 2
69--76
EN
Extremal problems for second order hyperbolic systems with multiple time-varying lags are presented. An optimal boundary control problem for distributed hyperbolic systems with boundary conditions involving multiple time-varying lags is solved. The time horizon is fixed. Making use of Dubovitski-Milyutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.
PL
Zaprezentowano ekstremalne problemy dla systemów hiperbolicznych z wielokrotnymi zmiennymi opóźnieniami czasowymi. Rozwiązano problem optymalnego sterowania brzegowego dla systemów hiperbolicznych drugiego rzędu, w których wielokrotne zmienne opóźnienia czasowe występują w warunkach brzegowych typu Neumanna. Tego rodzaju równania stanowią w liniowym przybliżeniu uniwersalny model matematyczny procesów fizycznych, w których ma miejsce przesyłanie sygnałów na odległość w liniach długich typu elektrycznego, hydraulicznego i innych. Korzystając z metody Dubowickiego-Milutina wyprowadzono warunki konieczne i wystarczające optymalności dla problemu liniowo-kwadratowego.
2
Content available Extremal problems for second order hyperbolic systems involving multiple time delays
Kowalewski Adam
Archives of Control Sciences
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2023
|
Vol. 33, No. 1
101--126
EN
Extremal problems for multiple time delay hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which multiple time delays appear both in the state equations and in the Neumann boundary conditions are solved. The time horizon is fixed. Making use of Dubovicki-Milutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.
3
Content available Extremal Problems for Infinite Order Parabolic Systems with Boundary Conditions Involving Integral Time Lags
Kowalewski Adam, Miśkowicz Marek
Pomiary Automatyka Robotyka
|
2022
|
R. 26, nr 4
37--42
EN
Extremal problems for integral time lag infinite order parabolic systems are studied in the paper. An optimal boundary control problem for distributed infinite order parabolic systems in which integral time lags appear in the Neumann boundary conditions is solved. Such equations constitute in a linear approximation a universal mathematical model for many diffusion processes (e.g., modeling and control of heat transfer processes). The time horizon is fixed. Using the Dubovicki-Milutin framework, the necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance indexes and constrained control are derived.
PL
Zaprezentowano ekstremalne problemy dla systemów parabolicznych nieskończonego rzędu z całkowymi opóźnieniami czasowymi. Rozwiązano problem optymalnego sterowania brzegowego dla systemów parabolicznych nieskończonego rzędu, w których całkowe opóźnienia czasowe występują w warunkach brzegowych Neumanna. Tego rodzaju równania stanowią w liniowym przybliżeniu uniwersalny model matematyczny dla procesów dyfuzyjnych. Korzystając z metody Dubowickiego-Milutina wyprowadzono warunki konieczne i wystarczające optymalności dla problemu liniowo-kwadratowego.
4
Content available Extremal problems for parabolic systems with time-varying lags
Kowalewski A.
Archives of Control Sciences
|
2018
|
Vol. 28, No. 1
89--104
EN
Extremal problems for parabolic systems with time-varying lags are presented. An optimal boundary control problem for parabolic systems in which time-varying lags appear in the state equations and in the boundary conditions simultaneously is solved. The time horizon is fixed. Making use of Dubovicki-Milutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.
5
Content available An adaptive control scheme for hyperbolic partial differential equation system (drilling system) with unknown coefficient
Farahani H. S., Telabi H. A., Baghermenhaj M.
Archives of Control Sciences
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2017
|
Vol. 27, no. 1
63--76
EN
The adaptive boundary stabilization is investigated for a class of systems described by second-order hyperbolic PDEs with unknown coefficient. The proposed control scheme only utilizes measurement on top boundary and assume anti-damping dynamics on the opposite boundary which is the main feature of our work. To cope with the lack of full state measurements, we introduce Riemann variables which allow us reformulate the second-order in time hyperbolic PDE as a system with linear input-delay dynamics. Then, the infinite-dimensional time-delay tools are employed to design the controller. Simulation results which applied on mathematical model of drilling system are given to demonstrate the effectiveness of the proposed control approach.
6
Content available Sequential optimization for semilinear divergent hyperbolic equation with a boundary control and state inequality constraint
Gavrilov V. S., Sumin M. I.
Control and Cybernetics
|
2014
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Vol. 43, no. 2
183--226
EN
An optimal control problem with a state constraint of inequality type and with dynamics described by a semilinear hyperbolic equation in divergence form with the non-homogeneous boundary condition of the third kind is considered. The state constraint contains a functional parameter that belongs to the class of continuous functions and occurs as an additive term. We study the properties of solutions of linear hyperbolic equations in divergence form with measures in the original data and compute the first variations of functionals on the basis of a so-called two-parameter needle variation of controls. We consider the necessary conditions for minimizing sequences in an optimal control problem with a pointwise in time state constraint of inequality type and with dynamics described by a semilinear hyperbolic equation in divergence form with the non-homogeneous boundary condition of the third kind. For the parametric optimization problem, we also consider regularity and normality conditions stipulated by the differential properties of its value function.
7
Content available Optimal boundary control problems of retarded parabolic systems
Kowalewski A., Krakowiak A.
Archives of Control Sciences
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2013
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Vol. 23, no. 3
261--279
EN
Optimal boundary control problems of retarded parabolic systems are presented. Necessary and sufficient conditions of optimality are derived for the Neumann problem. A simple example of application is also presented.
8
Content available Dirichlet control problems in smooth and nonsmooth convex plain domains
Casas E., Mateos M.
Control and Cybernetics
|
2011
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Vol. 40, no 4
931-955
EN
In this paper we collect some results about boundary Dirichlet control problems governed by linear partial differential equations. Some differences are found between problems posed on polygonal domains or smooth domains. In polygonal domains some difficulties arise in the corners, where the optimal control is forced to take a value which is independent of the data of the problem. The use of some Sobolev norm of the control in the cost functional, as suggested in the specialized literature as an alternative to the L2 norm, allows to avoid this strange behavior. Here, we propose a new method to avoid this undesirable behavior of the optimal control, consisting in considering a discrete perturbation of the cost functional by using a finite number of controls concentrated around the corners. In curved domains, the numerical approximation of the problem requires the approximation of the domain Ω typically by a polygonal domain Ωh, this introduces some difficulties in comparing the continuous and the discrete controls because of their definition on different domains Γ and Γh, respectively. We complete the existing recent analysis of these problems by proving the error estimates for a full discretization of the control problem. Finally, some numerical results are provided to compare the different alternatives and to confirm the theoretical predictions.
9
Content available Boundary control of retarded parabolic systems
Krakowiak A.
Control and Cybernetics
|
2011
|
Vol. 40, no 1
9-23
EN
Optimal boundary control problems for distributed parabolic systems in which retarded arguments appear in the integral form in the state equations are presented. Necessary and sufficient conditions of optimality are derived for the non-homogeneous Dirichlet problem. A simple example of application is also provided.
10
Content available remote On the regularization error of state constrained Neumann control problems
Krumbiegel K., Rosch A.
Control and Cybernetics
|
2008
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Vol. 37, no 2
369-392
EN
A linear elliptic optimal control problem with point-wise state constraints in the interior of the domain is considered. Furthermore, the control is given on the boundary with associated constraints. An artificial distributed control is introduced in the cost functional, in the state equation and in the state constraints. Since there are no control constraints for the artificial control, efficient numerical methods can be easily established. Based on a possible violation of the pure pointwise state constraints, an error estimate for the regularization error is derived. The theoretical results are illustrated by numerical tests.
11
Content available Strong decay for one-dimensional wave equations with nonmonotone boundary damping
Pierre M., Vancostenoble J.
Control and Cybernetics
|
2000
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Vol. 29, no 2
473-484
EN
This paper is a contribution to the following question : consider the classical wave equation damped by a nonlinear feedback control which is only assumed to decrease the energy. Then, do solutions to the perturbed system still exist for all time? Does strong stability occur in the sense that the energy tends to zero as time tends to infinity? We prove here that the answer to both questions is positive in the specific case of the one-dimensional wave equation damped by boundary controls which are functions of the observed velocity. The main point is that no monotonicity assumption is made on the damping term.
12
Content available Strong stability of a model of an overhead crane
Conrad F., Mifdal A.
Control and Cybernetics
|
1998
|
Vol. 27, no 3
363-374
EN
In this paper, a strong stability result is given for a model of an overhead crane which consists of a motorized platform moving along an horizontal beam with a flexible cable, holding a load mass M. A non uniform stability result is also shown.
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