We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and ...contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals 0
have a turnpike structure in the following sense: For large
the contribution to the objective functional that comes from the subinterval
, i.e., from the second half of the time interval 0
, is at most of the order 1
. More generally, the result holds for subintervals of the form
, where
(0, 1
2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in
with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.
The paper provides strong convergence of solutions to a sequence of linear-quadratic (LQ) optimization problems defined in an abstract functional framework. Each problem is accompanied by the ...constraint of reaching a given target within a prescribed precision. We show that the problems are well-posed and characterize their solutions. The main result provides the conditions under which these solutions converge to the minimizer of the limit problem. The generality of the result allows its application to a wide range of problems: elliptic, parabolic, control ones, etc. The examples presented in the paper consider optimal approximate controls of the heat equation and optimal approximate solutions to the Poisson equation.
We prove existence of solutions to Cauchy problem for scalar conservation laws with non-degenerate discontinuous flux $$ \partial_t u+ \hbox{div}f(t,\mathbf{x},u)=s(t,\mathbf{x},u), \quad t\geq 0, ...\mathbf{x}\in \mathbb{R}^d, $$ where for every $(t,\mathbf{x})\in \mathbb{R}^+\times \mathbb{R}$, the flux $f(t,\mathbf{x},\cdot) \in \hbox{Lip}(\mathbb{R};\mathbb{R}^d)$ and $\partial_\lambda f \in L^r(\mathbb{R}^+\times \mathbb{R}^d\times \mathbb{R})$, additionally satisfying $\max_{|\lambda| \leq M} f(\cdot,\cdot,\lambda) \in L^r(\mathbb{R}^+\times \mathbb{R}^d)$, for some $r>1$ and every $M>0$, and, for every $\lambda \in \mathbb{R}$, $\hbox{div}_{(t,\mathbf{x})} f(\cdot,\cdot,\lambda) \in \mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d)$ where $\mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d)$ is the space of Radon measures. Moreover, the function s is measurable and both f and s satisfy certain growth rate assumptions with respect to $\lambda$. The result is obtained by means of the H-measures.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
We analyse the turnpike properties for a general, infinite dimensional, linear–quadratic (LQ) optimal control problem, in both the deterministic and the parameter dependent case. The novelty of the ...paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for optimal averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result that extends the averaged controllability approach to optimal control problems In both cases, the results do not require assumptions on stabilizability and detectability, which are most commonly used in the study of turnpike phenomena.
This paper is concerned with the eigenvalue decay of solution operators to operator Lyapunov equations, a relevant topic in the context of model reduction for parabolic control problems. We mainly ...focus on the Gramian operator, which arises in the context of control and observation of heat processes in infinite time, which is normally the first step towards observations in a finite time horizon.
By improving existing energy and observability estimates for parabolic equations, we obtain both upper and lower bounds on the convergence rate of the eigenvalues of the Gramian operator towards zero. Both bounds follow the same polynomial decay rate, up to a multiplicative constant, which ensures their optimality. This confirms the slow decay of the eigenvalues and limits the efficiency of model reduction. The theoretical findings are supported by numerical results.
The aim of this paper is to provide an efficient method for solving a family of parameter dependent, algebraic Lyapunov equations in an infinite dimensional setting. Our analysis is based on previous ...work on reduced modeling and (weak) greedy algorithms for parameter dependent PDEs and abstract equations in Banach spaces. The major contribution is threefold. Firstly, the problem is resolved in an infinite dimensional setting, thus enabling applications to PDEs and equations governed by unbounded operators. Secondly, we demonstrate the boundedness and coercive properties of the Lyapunov operator in appropriate functional spaces. Lastly, the method is applied to the control theory, enabling rapid construction of approximate control functions for a wide range of control problems.
We analyse the problem of controllability for parameter dependent linear finite-dimensional systems. The goal is to identify the most distinguished realisations of those parameters so to better ...describe or approximate the whole range of controls. We adapt recent results on greedy and weak greedy algorithms for parameter dependent PDEs or, more generally, abstract equations in Banach spaces. Our results lead to optimal approximation procedures that, in particular, perform better than simply sampling the parameter-space to compute the controls for each of the parameter values. We apply these results for the approximate control of finite-difference approximations of the heat and the wave equation. The numerical experiments confirm the efficiency of the methods and show that the number of weak-greedy samplings that are required is particularly low when dealing with heat-like equations, because of the intrinsic dissipativity that the model introduces for high frequencies.
We demonstrate the stability of observability estimates for solutions to wave and Schrödinger equations subjected to additive perturbations. This work generalises recent averaged ...observability/control results by allowing for systems consisting of operators of different types. We also consider the simultaneous observability problem by which one tries to estimate the energy of each component of a system under consideration. Our analysis relies on microlocal defect tools, in particular on standard H-measures when the main system dynamic is governed by the wave operator, and parabolic H-measures in the case of the Schrödinger operator.
We analyze the problem of averaged observability and control of wave equations.
This topic is motivated by the control of parameter-dependent systems of wave equations. We look for controls ensuring ...the controllability of the averages of the states with respect to the parameter. This turns out to be equivalent to the problem of averaged observation in which one aims at recovering the energy of the initial data of the adjoint system by measurements done on its averages, under the assumption that the initial data of all the components of the adjoint system coincide.
The problem under consideration is weaker than the classical notion of simultaneous observation and control.
The method of proof uses propagation arguments based on H-measures or microlocal defect measures that reduce the problem to non-standard unique-continuation issues.
Using transmutation techniques, we also derive some results on the averaged observation and control of parameter-dependent heat equations.
On étudie le problème de l'observation et contrôle en moyenne d'équations des ondes.
Ce sujet est motivé par le contrôle d'équations des ondes dépendant de paramètres. On s'intéresse à la contrôlabilité des moyennes des états par rapport aux paramètres. Ceci équivaut au problème de l'observation des états adjoints dépendant des paramètres, mais tous avec les mêmes données initiales, en utilisant l'observation des moyennes.
Le problème considéré est plus faible que celui de la contrôlabilité ou de l'observabilité simultanées étudié antérieurement.
La méthode de démonstration s'appuie sur des arguments de propagation qui utilisent les H-mesures ou mesures de défaut microlocales, qui réduisent le problème à des questions de continuation unique.
En utilisant des arguments de transmutation, on obtient aussi quelques résultats pour le contrôle et l'observation en moyenne pour des équations paraboliques dépendant de paramètres.
A method is developed for analysing asymptotic behaviour of terms involving an arbitrary integer order powers of
functions by means of H-measures. It is applied to the small amplitude homogenisation ...problem for a stationary diffusion equation, in which coefficients are assumed to be analytic perturbations of a constant, enabling formulæ for higher order correction terms in a general, non-periodic setting. Explicit expressions in terms of Fourier coefficients are obtained under periodicity assumption. The method allows of its generalisation and application to the corresponding non-stationary equation, as well as to some other small amplitude homogenisation problems.
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