Large time control and turnpike properties for wave equations

Zuazua, Enrique

doi:10.1016/j.arcontrol.2017.04.002

Cited by 39 publications

(39 citation statements)

References 46 publications

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“…In Figure 9, we illustrate the efficiency of the steady control by plotting different curves representing the time evolution of the L 2 -norm of y solution to (40) with different initial datums and taking (41) as a control. To compare, we have computed the time-dependent solution y T (t) associated to the optimal control u T (t), obtained by minimizing (39), for the given parameter ν = √ 2 and y 0 (x) = 0. Note that we have split the time horizon in two different intervals for the sake of clarity.…”

Section: The Case C(x) ≥mentioning

confidence: 99%

“…For wave-like models, where solutions of the free dynamics are of oscillatory nature and do not enjoy the property of asymptotic simplification, is less clear. However, as shown in [31,39], the turnpike property still holds for wave-like models under suitable controllability assumptions. More precisely, except for an initial time layer [0, τ ] and a final one [0, T − τ ] during the rest of the time interval the solutions are exponentially close to the steady state ones.…”

Section: Additional Comments and Open Problemsmentioning

confidence: 99%

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Greedy optimal control for elliptic problems and its application to turnpike problems

2018

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In this paper, we deal with the approximation of optimal controls for parameter-dependent elliptic and parabolic equations. We adapt well-known results on greedy algorithms to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameterspace to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. To avoid this difficulty we employ the turnpike property for time evolution control problems that ensures the asymptotic simplification of optimal control problems for evolution equations towards the elliptic steady-state ones in long time horizons [0, T ]. The combination of the turnpike property and greedy methods allows us to develop efficient methods for the approximation of the parameter-dependent parabolic optimals too. We present various numerical experiments discussing the efficiency of our methodology and its application to turnpike control problems. Keywords parameterized PDEs • optimal control • turnpike property • greedy algorithms • elliptic equations • parabolic equations Mathematics Subject Classification (2000) 49J20 • 49K20 • 93C20 • 49N05 • 65K10

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Section: The Case C(x) ≥mentioning

confidence: 99%

Section: Additional Comments and Open Problemsmentioning

confidence: 99%

Greedy optimal control for elliptic problems and its application to turnpike problems

2018

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“…As discussed in [36], a key point to get the convergence of finite horizon optimal control problems as T tends to infinity is to establish the role played by observability estimates: even if the underlying system is unstable or oscillatory (see [44]) the turnpike property will hold. Here we have proved the observability estimate (2.2) for the adjoint system (2.1), hence, following the spirit of [36], we expect that the turnpike property holds likewise for (1.3).…”

Section: Turnpike Propertymentioning

confidence: 99%

Controllability of shadow reaction-diffusion systems

Hernández-Santamaría

Zuazua

2020

Journal of Differential Equations

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We study the null controllability of linear shadow models for reaction-diffusion systems arising as singular limits when the diffusivity of some of the components is very high. This leads to a coupled PDE-ODE system where one component solves a parabolic partial differential equation (PDE) and the other one an ordinary differential equation (ODE). This reduced system contains the essential dynamics of the original one. We analyze these shadow systems from a controllability perspective and prove two types of results. First, by employing Carleman inequalities, ODE arguments and regularity results, we prove that the null controllability of the shadow model holds. This result, together with the effectiveness of the controls of the shadow system to control the full original dynamics for large values of the diffusivity parameter, is then illustrated by numerical simulations. We also obtain a uniform Carleman estimate for the reaction-diffusion equations which allows to obtain the null control for the shadow system as a limit when the diffusivity tends to infinity in one of the equations. These results justify the efficiency of shadow systems not only for modeling the dynamics of reactiondiffusion system in the large diffusivity limit of one of the equations, but at the control level too.

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“…Proof: We again employ the eigen structure analysis (13). The tangent spaces of S and U at the origin are written as…”

Section: B the Ocp With State Variables Specified At The Terminal Timementioning

confidence: 99%

“…We, then, apply this inequality to an optimal control problems in which terminal states are not specified and the steady optimal solutions are not the origin as in [9], [11], [13], [14] and to an optimal control problem in which two terminal states are specified and the steady optimal point is the origin as in [3], [5], [10]. For both classes of problems, we employ a Dynamic Programming approach with Hamilton-Jacobi equations (HJEs).…”

Section: Introductionmentioning

confidence: 99%

The turnpike property in nonlinear optimal control — A geometric approach

Sakamoto

Pighin

Zuazua

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike occurs. The approach taken is geometric and gives insights for the behaviors of controlled trajectories, allowing us to find simpler proofs for existing results on the turnpike properties.

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Large time control and turnpike properties for wave equations

Cited by 39 publications

References 46 publications

Greedy optimal control for elliptic problems and its application to turnpike problems

Greedy optimal control for elliptic problems and its application to turnpike problems

Controllability of shadow reaction-diffusion systems

The turnpike property in nonlinear optimal control — A geometric approach

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