Error Estimates for Space-Time Discretization of Parabolic Time-Optimal Control Problems with Bang-Bang Controls

Bonifacius, Lucas; Pieper, Konstantin; Vexler, Boris

doi:10.1137/18m1213816

Cited by 11 publications

(12 citation statements)

References 31 publications

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confidence: 98%

“…Since the jump set of the optimal control is contained in the set of global extreme points ofΦ, see Corollary 1, this assumption implies that the optimal control admits only finitely many jumps, which is a rather typical situation in practice. In addition, it ensures that the adjoint state has nonvanishing first derivative near the global extreme points ofΦ, which is closely related to assumptions used to derive error estimates for bang-bang control problems, see, e.g., [7,20,24].Starting from possibly suboptimal error estimates for the state and adjoint state and incorporating the structural assumption, we are able to derive an error estimate for the controls in L 1 for both variational control discretization, where the order of the error is O(h 2 ), and piecewise constant control discretization, where we obtain O(h). Moreover, we provide numerical experiments which indicate that the established error estimates are optimal.…”

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confidence: 99%

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Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

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Here, u ∈ V := H 1 0 (Ω) is the state and q ∈ Q := BV (Ω) is the control, where BV (Ω) denotes the space of functions of bounded variation (BV) on the interval Ω := (0, 1). The operator A is elliptic and α is a positive real number. The two finite element schemes that will be analyzed are identical in regard to the discretization of state and adjoint state, but they differ in the treatment of the control. In the variational discretization the control is not discretized, while in the second scheme the control is discretized by piecewise constant functions.The significance of the above control problem is given by the use of the BVseminorm q ′ M(Ω) in the objective. This favors piecewise constant controls with only a limited number of jumps, which makes this problem type interesting in many practical applications. The precise functional analytic setting will be provided in the next section.Optimal control problems with BV-controls defined in one space dimension are strongly related to control problems with measures as controls. Both BV optimal control problems and optimal control problems with measures have attracted significant research interest in the recent past, see, e.g. [8,13,14,17,22,23] for the former and [11,12,15,16,29,30] for the latter.Error estimates for PDE-constrained optimal control problems involving measures have been presented in [11,30,31,34,35]. For error estimates of further sparsity promoting optimal control problems with PDEs see for example [19,30]. The literature on error estimates for optimal control problems with controls in BV is rather limited. We are only aware of [14,18]. Error estimates and numerical analysis for inverse problems involving BV-functions are studied in [5,6]. Related discussion of ODE-constrained control problems involving discontinuous functions and their numerical analysis can be found in, e.g., [1,2,10,21,25,26,37,38].The main difficulty in deriving error estimates for the above problem is given by the fact that it lacks certain coercivity properties that are usually employed to obtain error estimates for the controls, for instance by suitably testing the first order necessary optimality conditions. Hence, only error estimates for the state and the adjoint state can be proven in a rather direct manner; these are, however, suboptimal. To obtain an error estimate for the control and also to improve the error estimates for state and adjoint state, we make use of a structural assumption on the Lagrange multiplierΦ arising from the convex subdifferential of the term q ′ M(Ω) . Specifically, we assume thatΦ, which is a C 2 function inΩ, has only finitely many global extreme points and that it exhibits quadratic growth near those points (i.e.,Φ ′′ = 0 near those points; see Assumption 4.4 and Assumption 4.5). Since the jump set of the optimal control is contained in the set of global extreme points ofΦ, see Corollary 1, this assumption implies that the optimal control admits only finitely many jumps, which is a rather typical situation in practice. In addition, it ensures ...

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confidence: 99%

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

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“…A second novelty of this paper is the proof of error estimates for the difference between the discrete and continuous controls in the framework of parabolic control problems with bang-bang controls. As far as we know, the only results in this direction are obtained for linear state equations: in [17], a quadratic convergence order is obtained for the error with respect to the time step size in the case of bang-bang controls using a variational discretization; in [3], the authors study a time optimal control problem and obtain results similar to ours (compare [3, Table 1.1] and Theorem 6.5). For results in the case of the control of elliptic equations the reader can consult [18] for linear equations with variational discretization and [15] for semilinear equations with full discretization.…”

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confidence: 72%

“…For specific examples, we just need to defineφ andū. With these choices, we define 3 . We have that (ū,ȳ,φ) satisfies the first order optimality conditions (3.1a)-(3.1c).…”

Section: Numerical Experimentmentioning

confidence: 99%

Error Estimates for Semilinear Parabolic Control Problems in the Absence of Tikhonov Term

Casas¹,

Mateos²,

Rösch³

2019

SIAM J. Control Optim.

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In this paper, we analyze optimal control problems of semilinear parabolic equations, where the controls are distributed and depend only on time. Box constraints for the controls are imposed and the cost functional does not involve the control itself, only the associated state. We prove second order optimality conditions for local strong minimizers, which are used to derive error estimates in the numerical approximation. First we estimate the difference between the discrete and continuous optimal states. In the last part, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states.

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“…The tracking differentiator (TD) via the time optimal control (TOC) was first proposed by Han (1) . However, this continuous-time time-optimal solution (i.e., bang-bang control) in designing the TD introduces considerable numerical errors in a discrete-time implementation (2) . This is because, in the bang-bang control, the control signal switches frequently between two extreme values around the switching curve, particularly the origin (3) .…”

Section: Introductionmentioning

confidence: 99%

Discrete-Time Optimal Control of Double Integrators and its Application in Maglev Train

Zhang

Xie

et al. 2022

IEEJ Journal IA

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Non-member Zhiqiang Long * 5 Non-member As an alternative to bang-bang control, a time optimal control (TOC) algorithm for discrete-time systems was first reported by Han (1) . This algorithm not only acts as a noise-tolerant tracking differentiator (TD) to avoid setpoint jumps in control processes, but also has wide applications in the design of controllers and observers. However, determination of the real-time state position on the phase plane involves complex boundary transformations, which renders this algorithm impractical for some engineering applications. This paper proposes a methodology for discrete-time optimal control (DTOC) of double integrators with disturbances. The closed-form solution with lower computational burden can be easily extended to general second-order systems. Further, in consideration of the inevitable disturbances in the systems, a rigorous and full-convergence proof is presented for the proposed algorithm. The results show finite-time and fast convergence as well as provide the ultimate stable attraction regions for the system states. Examples and experiments are also presented to demonstrate the effectiveness of the proposed algorithm for solving a signal processing problem in a maglev train.

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Error Estimates for Space-Time Discretization of Parabolic Time-Optimal Control Problems with Bang-Bang Controls

Cited by 11 publications

References 31 publications

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Error Estimates for Semilinear Parabolic Control Problems in the Absence of Tikhonov Term

Discrete-Time Optimal Control of Double Integrators and its Application in Maglev Train

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