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1
Minimum energy control of degenerate Cauchy problem with skew-Hermitian pencil
Benaoued Meriem, Bouagada Djillali
Journal of Applied Analysis
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2022
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Vol. 28, nr 2
251--261
EN
The present research paper deals with the effectiveness of the control of an infinite-dimensional degenerate Cauchy problem with variable operator coefficients, skew-Hermitian pencil and bounded input condition. This study explores the minimum energy control problem. The investigation follows a set of methods to examine the procedure for developing a new result to solve the problem. Indeed, by the use of decomposition transformation of the considered system and the application of the Gramian operator, the formula of the process for controlling the system with minimum energy is obtained. Afterwards, a procedure to compute the optimal input for minimizing the performance index is then proposed. In a nutshell, the obtained results indicate that optimal control for minimizing the performance index ensures the solution of the minimum energy control of an infinite-dimensional degenerate Cauchy problem.
2
Content available Controllability of time varying semilinear non-instantaneous impulsive systems with delay, and nonlocal conditions
Cabada Dalia, Garcia Katherine, Guevara Cristi, Leiva Hugo
Archives of Control Sciences
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2022
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Vol. 32, No. 2
335--357
EN
In this paper we prove the exact controllability of a time varying semilinear system considering non-instantaneous impulses, delay, and nonlocal conditions occurring simultaneously. It is done by using the Rothe’s fixed point theorem together with some sub-linear conditions on the nonlinear term, the impulsive functions, and the function describing the nonlocal conditions. Furthermore, a control steering the semilinear system from an initial state to a final state is exhibited.
3
Content available On properties of minimizers of a control problem with time-distributed functional related to parabolic equations
Astashova I. V., Filinovskiy A. V.
Opuscula Mathematica
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2019
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Vol. 39, no. 5
595--609
EN
We consider a control problem given by a mathematical model of the temperature control in industrial hothouses. The model is based on one-dimensional parabolic equations with variable coefficients. The optimal control is defined as a minimizer of a quadratic cost functional. We describe qualitative properties of this minimizer, study the structure of the set of accessible temperature functions, and prove the dense controllability for some set of control functions.
4
Content available Exact controllability of a string to rest with a moving boundary
Gugat Martin
Control and Cybernetics
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2019
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Vol. 48, No. 1
69--87
EN
We consider the problem of steering a finite string to the zero state in finite time from a given initial state by controlling the state at one boundary point while the other boundary point moves. As a possible application we have in mind the optimal control of a mining elevator, where the length of the string changes during the transportation process. During the transportation process, oscillations of the elevator-cable can occur that can be damped in this way. We present an exact controllability result for Dirichlet boundary control at the fixed end of the string that states that there exist exact controls for which the oscillations vanish after finite time. For the result we assume that the movements are Lipschitz continuous with a Lipschitz constant, whose absolute value is smaller than the wave speed. In the result, we present the minimal time, for which exact controllability holds, this time depending on the movement of the boundary point. Our results are based upon travelling wave solutions. We present a representation of the set of successful controls that steer the system to rest after finite time as the solution set of two point-wise equalities. This allows for a transformation of the optimal control problem to a form where no partial differential equation appears. This representation enables interesting insights into the structure of the successful controls. For example, exact bang-bang controls can only exist if the initial state is a simple function and the initial velocity is zero.
5
Content available Distributed controllability of one-dimensional heat equation in unbounded domains: The Green's function approach
Khurshudyan Asatur Zh.
Archives of Control Sciences
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2019
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Vol. 29, No. 1
57--71
EN
We derive exact and approximate controllability conditions for the linear one-dimensional heat equation in an infinite and a semi-infinite domains. The control is carried out by means of the time-dependent intensity of a point heat source localized at an internal (finite) point of the domain. By the Green’s function approach and the method of heuristic determination of resolving controls, exact controllability analysis is reduced to an infinite system of linear algebraic equations, the regularity of which is sufficient for the existence of exactly resolvable controls. In the case of a semi-infinite domain, as the source approaches the boundary, a lack of L2-null-controllability occurs, which is observed earlier by Micu and Zuazua. On the other hand, in the case of infinite domain, sufficient conditions for the regularity of the reduced infinite system of equations are derived in terms of control time, initial and terminal temperatures. A sufficient condition on the control time, heat source concentration point and initial and terminal temperatures is derived for the existence of approximately resolving controls. In the particular case of a semi-infinite domain when the heat source approaches the boundary, a sufficient condition on the control time and initial temperature providing approximate controllability with required precision is derived.
6
Content available On controllability of second order dynamical systems – a survey
Klamka J., Wyrwał J., Zawiski R.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2017
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Vol. 65, nr 3
279--295
EN
The paper presents a survey of recent results in the area of controllability of second order dynamical systems. Controllability problem for finite and infinite dimensional, linear, semilinear, deterministic and stochastic dynamical systems (with delays and undelayed) is taken into consideration. Different types of controllability are discussed.
7
Content available Controllability of semilinear systems with fixed delay in control
Kumar S., Sukavanam N.
Opuscula Mathematica
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2015
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Vol. 35, no. 1
71--83
EN
In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assump­tions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given.
8
Content available remote Exact controllability for the semilinear string equation in non cylindrical domains
Araruna F. D., Antunes G. O., Medeiros L. A.
Control and Cybernetics
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2004
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Vol. 33, no 2
237-257
EN
In this paper, we investigate the exact controllability for a mixed problem for the equation u^n - [...] + f(u) = 0 in a non cylindrical domain. This model, without the resistance represented for f(u), is a linearization of Kirchhoff's equation for small vibrations of a stretched elastic string when the ends are variables, see Medeiros, Limaco, Menezes (2002). We employ a variant, due to Zuazua (1990b), of the Hilbert Uniqueness Method (HUM), idealized by Lions (1988a, b).
9
Content available remote Exact Controllability of an Elastic Membrane Coupled With a Potential Fluid
Hansen S.
International Journal of Applied Mathematics and Computer Science
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2001
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Vol. 11, no 6
1231-1248
EN
We consider the problem of boundary control of an elastic system with coupling to a potential equation. The potential equation represents the linearized motions of an incompressible inviscid fluid in a cavity bounded in part by an elastic membrane. Sufficient control is placed on a portion of the elastic membrane to insure that the uncoupled membrane is exactly controllable. The main result is that if the density of the fluid is sufficiently small, then the coupled system is exactly controllable.
10
Content available remote Linear Quadratic Control Problem With Fixed Final State for Discrete-Time Distributed Systems
Chraibi L., Karrakchou J., Ouansafi A., Rachik M.
International Journal of Applied Mathematics and Computer Science
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2000
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Vol. 10, no 3
517-535
EN
The problem considered is that of minimizing a quadratic cost functional for a discrete distributed system with fixed initial and final states. It is shown that under suitable controllability assumptions, there is a close relationship between this problem and that of exact controllability with minimization of a time-varying energy criterion. The HUM technique is then extended to treat the exact controllability problem in the time-varying case and applied to provide an explicit form for the optimal control and the optimal cost.
11
Content available Inverse/observability estimates for Schroedinger equations with variable coefficients
Triggiani R., Yao P.
Control and Cybernetics
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1999
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Vol. 28, no 3
627-664
EN
We consider a general Schroedinger equation defined on an open bounded domain [Omega is a subset of R^n] with variable coefficients in both the elliptic principal part and in the first-order terms as well. At first, no boundary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H[sup l](Omega)-energy at time t = T, or at time t = 0, is dominated by the L[sub2](Sigma)-norms of the boundary traces [...] and w[sub t] modulo an interior lower-order term. Once homogeneous B.C. are imposed, our results yield - under a uniqueness theorem, needed to absorb the lower order term - continuous observability estimates for both the Dirichlet and Neumann case, with an arbitrarily short observability time ; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neuman case. In contrast to existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in [Omega is a subset of R^n] to a problem on an appropriate Riemannian manifold (determined by the coefficients of the principal part), where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the va.riable first-order (energy level) terms. In our third step, we employ micro-local analysis yielding a sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the Neumann case.
12
Content available Domain decomposition in exact controllability of second order hyperbolic systems on 1-d networks
Lagnese J.
Control and Cybernetics
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1999
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Vol. 28, no 3
531-556
EN
This paper is concerned with domain decomposition in exact controllability of a class of linear second order hyperbolic systems on one-dimensional graphs in [R^3] that in particular serve as descriptive models of the dynamics of various multi-link structures consisting of one-dimensional elements, such as networks of Timoshenko beams in [R^3]. We first consider a standard unconstrained optimal control problem in which the cost functional penalizes the deviation of the final state of the global problem from a given target state. A convergent domain decomposition for the optimality system associated with this problem was recently given by G. Leugering. This decomposition depends on the penalty parameter. On each edge of the graph and at each iteration level the local problem is itself the optimality system associated with an unconstrained optimal control problem in which the cost functional penalizes the deviation of the final state of the particular edge from the target state for that edge. The main purpose of this paper is to show that at each iteration level and on each edge the local optimality system converges as the penalty parameter approaches its limit and that the limit system is a domain decomposition for the problem of norm minimum exact control to the target state.
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