Switching control

Enrique Zuazua

Displaying similar documents to “Switching control”

Stability and sensitivity analysis for optimal control problems with control-state constraints

Kazimierz Malanowski

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A family of parameter dependent optimal control problems ${(O)}_{h}$ with smooth data for nonlinear ODEs is considered. The problems are subject to pointwise mixed control-state constraints. It is assumed that, for a reference value h₀ of the parameter, a solution of ${(O)}_{h ₀}$ exists. It is shown that if (i) independence, controllability and coercivity conditions are satisfied at the reference solution, then (ii) for each h from a neighborhood of h₀, a locally unique solution to ${(O)}_{h}$ and the associated Lagrange...

Synthesis of optimal control for nonlinear third order systems

Wiesław Szwiec

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The paper is devoted to a certain problem in the theory of synthesis of optimal control. The fundamental results of research in the synthesis of optimal control for second order systems are given in [1], [2], [5], [10]. In particular, in [2] and [5] some specific nonlinear systems are investigated, namely the systems related with the differential equations of the forms̈ + F(s,ṡ) = u or s̈ + F(s,ṡ,u) = 0where u is a control parameter in the interval [—1,1]. Some results concerning third...

Limiting average cost control problems in a class of discrete-time stochastic systems

Nadine Hilgert, Onesimo Hernández-Lerma (2001)

Applicationes Mathematicae

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We consider a class of $ℝ^{d}$ -valued stochastic control systems, with possibly unbounded costs. The systems evolve according to a discrete-time equation $x_{t + 1} = G ₙ (x_{t}, a_{t}) + ξ_{t}$ (t = 0,1,... ), for each fixed n = 0,1,..., where the $ξ_{t}$ are i.i.d. random vectors, and the Gₙ are given functions converging pointwise to some function $G_{\infty}$ as n → ∞. Under suitable hypotheses, our main results state the existence of stationary control policies that are expected average cost (EAC) optimal and sample path average cost (SPAC)...

Distances to convex sets

Antonio S. Granero, Marcos Sánchez (2007)

Studia Mathematica

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If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d ̂ ({\bar{c o}}^{w *} (K), C) : = s u p i n f | | k - c | | : c \in C : k \in {\bar{c o}}^{w *} (K)$ from ${\bar{c o}}^{w *} (K)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d ̂ ({\bar{c o}}^{w *} (K), C) \leq M d ̂ (K, C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically...

Model following control system with time delays

Dazhong Wang, Shujing Wu, Wei Zhang, Guoqiang Wang, Fei Wu, Shigenori Okubo (2016)

Kybernetika

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Design of model following control system (MFCS) for nonlinear system with time delays and disturbances is discussed. In this paper, the method of MFCS will be extended to nonlinear system with time delays. We set the nonlinear part $f (v (t))$ of the controlled object as $| | f (v (t)) | | \leq α + {β | | v (t) | |}^{γ}$ , and show the bounded of internal states by separating the nonlinear part into $γ \geq 0$ . Some preliminary numerical simulations are provided to demonstrate the effectiveness of the proposed method.

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D \subset ℝ^{d}$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $𝒫$ . We associate to Problem $𝒫$ an optimal control problem, denoted by $𝒬$ . Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\tilde{K}$ , we provide results concerning the well-posedness...

Linearization techniques for $𝕃^{\infty}$ See PDF-control problems and dynamic programming principles in classical and $𝕃^{\infty}$ See PDF-control problems

Dan Goreac, Oana-Silvia Serea (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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The aim of the paper is to provide a linearization approach to the $𝕃^{\infty}$ See PDF-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $𝕃^{p}$ See PDF approach and the associated linear formulations. This seems to be the most appropriate tool for treating $𝕃^{\infty}$ See PDF problems in continuous and...

An observability estimate for parabolic equations from a measurable set in time and its applications

Kim Dang Phung, Gengsheng Wang (2013)

Journal of the European Mathematical Society

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This paper presents a new observability estimate for parabolic equations in $Ω \times (0, T)$ , where $Ω$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $Ω$ and a subset of positive measure in $(0, T)$ . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

Transportation flow problems with Radon measure variables

Marcus Wagner (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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For a multidimensional control problem ${(P)}_{K}$ involving controls $u \in L_{\infty}$ , we construct a dual problem ${(D)}_{K}$ in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of $(L_{\infty}) *$ . For this purpose, we add to ${(P)}_{K}$ a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

Robust observer-based finite-time $H_{\infty}$ control designs for discrete nonlinear systems with time-varying delay

Yali Dong, Huimin Wang, Mengxiao Deng (2021)

Kybernetika

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This paper investigates the problem of observer-based finite-time $H_{\infty}$ control for the uncertain discrete-time systems with nonlinear perturbations and time-varying delay. The Luenberger observer is designed to measure the system state. The observer-based controller is constructed. By constructing an appropriated Lyapunov-.Krasovskii functional, sufficient conditions are derived to ensure the resulting closed-loop system is $H_{\infty}$ finite-time bounded via observer-based control. The observer-based...

Cyclic Type Fixed Point Results in 2-Menger Spaces

Binayak S. Choudhury, Samir Kumar BHANDARI, Parbati SAHA (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we introduce generalized cyclic contractions through $r$ number of subsets of a probabilistic 2-metric space and establish two fixed point results for such contractions. In our first theorem we use the Hadzic type $t$ -norm. In another theorem we use a control function with minimum $t$ -norm. Our results generalizes some existing fixed point theorem in 2-Menger spaces. The results are supported with some examples.

Output feedback $H_{\infty}$ control of networked control systems based on two channel event-triggered mechanisms

Yanjun Shen, Zhenguo Li, Gang Yu (2021)

Kybernetika

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In this paper, we study dynamical output feedback $H_{\infty}$ control for networked control systems (NCSs) based on two channel event-triggered mechanisms, which are proposed on both sides of the sensor and the controller. The output feedback $H_{\infty}$ controller is constructed by taking random network-induced delays into consideration without data buffer units. The controlled plant and the output feedback controller are updated immediately by the sampled input and the sampled output, respectively. By...

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

Control for Schrödinger operators on 2-tori: rough potentials

Jean Bourgain, Nicolas Burq, Maciej Zworski (2013)

Journal of the European Mathematical Society

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For the Schrödinger equation, $(i \partial_{t} + \nabla) u = 0$ on a torus, an arbitrary non-empty open set $Ω$ provides control and observability of the solution: $∥u {|_{t = 0}∥}_{L^{2} (𝕋^{2})} \leq K_{T} {∥u∥}_{L^{2} ([0, T] \times Ω)}$ . We show that the same result remains true for $(i \partial_{t} + \nabla - V) u = 0$ where $V \in L^{2} (𝕋^{2})$ , and $𝕋^{2}$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V \in C (𝕋^{2})$ and conjectured for $V \in L^{\infty} (𝕋^{2})$ . The higher dimensional generalization remains open for $V \in L^{\infty} (𝕋^{n})$ .

On the stabilization problem for nonholonomic distributions

Ludovic Rifford, Emmanuel Trélat (2009)

Journal of the European Mathematical Society

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Let $M$ be a smooth connected complete manifold of dimension $n$ , and $Δ$ be a smooth nonholonomic distribution of rank $m \leq n$ on $M$ . We prove that if there exists a smooth Riemannian metric on1for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $Δ$ on $M$ . Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific...

Systems of Bellman Equations to Stochastic Differential Games with Discount Control

Alain Bensoussan, Jens Frehse (2008)

Bollettino dell'Unione Matematica Italiana

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We consider two dimensional diagonal elliptic systems $\Delta u+au=H(x,u,\nabla u)$ which arise from stochastic differential games with discount control. The Hamiltonians $H$ have quadratic growth in $\nabla u$ and a special structure which has notyet been covered by regularity theory. Without smallness condition on $H$ , the existence of a regular solution is established.

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...