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Convergence of minimal norm control problems of linear heat equations with time delay
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This paper studies an equivalence theorem for three different kinds of optimal control problems, which are optimal time control problems, optimal norm control problems, and optimal target control problems. The controlled systems in this paper are internally controlled linear heat equations with memory.
This paper studies a kind of minimal norm optimal control problem for a semilinear heat equation with impulse controls; it consists in finding an impulse control, which has the minimal norm among all controls steering solutions of the controlled system to a given target from an initial state in a fixed time interval. We will study the existence of optimal controls to this problem. Then, we will establish a nontrivial Pontryagin’s maximum principle of this problem.
There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming L¹-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For L¹-type objectives we obtain extremal controls of bang-bang type. © 2018 American Institute of Mathematical Sciences. All rights reserved.
In this work we study the asymptotic behavior of the solutions of a class of
abstract parabolic time optimal control problems when the generators converge,
in an appropriate sense, to a given strictly negative operator. Our main
application to PDEs systems concerns the behavior of optimal time and of the
associated optimal controls for parabolic equations with highly oscillating
coefficients, as we encounter in homogenization theory. Our main results assert
that, provided that the target is a closed ball centered at the origin and of
positive radius, the solutions of the time optimal control problems for the
systems with oscillating coefficients converge, in the usual norms, to the
solution of the corresponding problem for the homogenized system. In order to
prove our main theorem, we provide several new results, which could be of a
broader interest, on time and norm optimal control problems.
We prove the equivalence of the minimal time and minimal norm control prob-lems for heat equations on bounded smooth domains of the euclidean space with homogeneous Dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves. We consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time. As a consequence of this equivalence, using the well-known variational characterization of minimal norm controls, we establish necessary and sufficient conditions for the minimal time and the corresponding control.
We prove new results on the continuity properties and on reachable sets of the minimum time function associated with a linear dynamical system in a separable Hilbert space H. Part of the results are new also in the finite dimensional case. The regularity results are stated thanks to the result we obtain on the connection between the minimum time function and the minimum energy.
In this paper, we study a certain approximation property for a time optimal
control problem of the heat equation with -potential. We prove that
the optimal time and the optimal control to the same time optimal control
problem for the heat equation, where the potential has a small perturbation,
are close to those for the original problem. We also verify that for the heat
equation with a small perturbation in the potential, one can construct a new
time optimal control problem, which has the same target as that of the original
problem, but has a different control constraint bound from that of the original
problem, such that the new and the original problems share the same optimal
time, and meanwhile the optimal control of the new problem is close to that of
the original one. The main idea to approach such approximation is an
appropriate use of an equivalence theorem of minimal norm and minimal time
control problems for the heat equations under consideration. This theorem was
first established by G.Wang and E. Zuazua in [20] for the case where the
controlled system is an internally controlled heat equation without the
potential and the target is the origin of the state space.
This technical note shows the identifiability of a nonlinear delayed-differential model describing aircraft dynamics. This result is obtained by an original combination: An extension of the results of Grewal and Glover to delay systems and an application of linear delayed-differential model identifiability of Orlov et al. . It requires initial conditions corresponding to an equilibrium state, which is implemented during the experimentation.
The control parameterization method used together with the time-scaling transformation is an effective approach to approximating optimal control problems into optimal parameter selection problems when no time delays are involved. The approximate problems can then be solved by gradient-based optimization algorithms. However, the time-scaling transformation, which works well for optimizing variable switching times of the approximate piecewise constant/linear control functions obtained after the application of the control parameterization method, is not applicable to optimal control problems with time delays. In this paper, we consider a class of nonlinear optimal control problems with multiple time delays subject to canonical equality and inequality constraints. Our aim is to develop a novel transformation procedure that converts a given time-delay system into an equivalent system-defined on a new time horizon-in which the control switching times are fixed, but the dynamic system contains multiple variable time delays expressed in terms of the durations between the switching times for each of the approximate control functions in the original time horizon. On this basis, we show that an optimal control policy for the equivalent system can be obtained efficiently using gradient-based optimization techniques. This optimal control policy can then be used to determine the optimal switching times and optimal control variables for the original system. We conclude the paper by solving two example problems.
In a recent, related, paper, necessary conditions in the form of a Maximum Principle were derived for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions covered fixed end-time problems and, under additional hypotheses, free end-time problems. These conditions improved on previous conditions in the following respects. They provided the first fully non-smooth Pontryagin Maximum Principle for problems involving delays in both state and control variables, only special cases of which were previously available. They provide a strong version of the Weierstrass condition for general problems with possibly non-commensurate control delays, whereas the earlier literature does so only under structural assumptions about the dynamic constraint. They also provided a new ‘two-sided’ generalized transversality condition, associated with the optimal end-time. This paper provides an extension of the Pontryagin Maximum Principle of the earlier paper for time delay systems, to allow for the presence of a unilateral state constraint. The new results fully recover the necessary conditions of the earlier paper when the state constraint is absent, and therefore retain all their advantages but in a setting of greater generality.
In this paper, we consider an optimal control model governed by a system of delay differential equations representing an SIR model. We extend the model of Kaddar (2010) by incorporating the suitable controls. We consider two control strategies in the optimal control model, namely: the vaccination and treatment strategies. The model has three time delays that represent the incubation period, and the times taken by the vaccine and treatment to be effective. We derive the first-order necessary conditions for the optimal control and perform numerical simulations to show the effectiveness as well as the applicability of the model for different values of the time delays. These numerical simulations show that the model is more sensitive to the delays representing the incubation period and the treatment delay, whereas the delay associated with the vaccine is not significant.
The primary concern of this paper is the treatment of optimal control of heating processes on the boundary of the spatial domains in which the processes take place. Given an initial state of the temperature at the timet=0 and a final state at some timeT>0, the problem is considered to find a control on the boundary by which the initial state is transferred to the final state. If in addition the control is subject to a restriction in absolute value, the question is answered under which conditions restricted control is possible in minimum time and time-minimal controls are characterized by a “bang-bang-principle”.
As mathematical tool certain exponential moment problems are investigated and the results are applied.
In a recent paper we presented the first adaptive control design for an ODE system with a possibly large actuator delay of unknown length. We achieved global stability under full state feedback. In this paper we generalize the design to the situation where, besides the unknown delay value, the ODE also has unknown parameters, and where trajectory tracking (rather than equilibrium regulation) is pursued.
Time Optimal Control of Evolution Equations
- G S Wang
- L J Wang
- Y S Xu
- Y B Zhang
G. S. Wang, L. J. Wang, Y. S. Xu and Y. B. Zhang, Time Optimal Control of Evolution Equations, Progr. Nonlinear Differential Equations Appl., 92, Subser. Control,
Birkhäuser/Springer, Cham, 2018.