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

Kazimierz Malanowski

Displaying similar documents to “Stability and sensitivity analysis for optimal control problems with control-state constraints”

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)...

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...

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.

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.

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.

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...

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.

On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters

Aldo Bressan (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot)$ . However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot)$ , where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\Sigma$ , referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy...

On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters

Aldo Bressan (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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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 $V \in L^{\infty} (𝕋^{n})$ .

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...

The covering semigroup of invariant control systems on Lie groups

Víctor Ayala, Eyüp Kizil (2016)

Kybernetika

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It is well known that the class of invariant control systems is really relevant both from theoretical and practical point of view. This work was an attempt to connect an invariant systems on a Lie group $G$ with its covering space. Furthermore, to obtain algebraic properties of this set. Let $G$ be a Lie group with identity $e$ and $Σ \subset 𝔤$ a cone in the Lie algebra $𝔤$ of $G$ that satisfies the Lie algebra rank condition. We use a formalism developed by Sussmann, to obtain an algebraic structure on...

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...

Stability estimating in optimal stopping problem

Elena Zaitseva (2008)

Kybernetika

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We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space $X$ . It is supposed that an unknown transition probability $p (\cdot | x)$ , $x \in X$ , is approximated by the transition probability $\tilde{p} (\cdot | x)$ , $x \in X$ , and the stopping rule ${\tilde{τ}}_{*}$ , optimal for $\tilde{p}$ , is applied to the process governed by $p$ . We found an upper bound for the difference between the total expected cost, resulting when applying ${\tilde{τ}}_{*}$ , and the minimal total expected cost. The bound given is a constant times ${sup}_{x \in X} ∥ p (\cdot | x) - \tilde{p} (\cdot | x) ∥$ , where $∥ \cdot ∥$ is the...

Time-varying Markov decision processes with state-action-dependent discount factors and unbounded costs

Beatris A. Escobedo-Trujillo, Carmen G. Higuera-Chan (2019)

Kybernetika

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In this paper we are concerned with a class of time-varying discounted Markov decision models $ℳ_{n}$ with unbounded costs $c_{n}$ and state-action dependent discount factors. Specifically we study controlled systems whose state process evolves according to the equation $x_{n + 1} = G_{n} (x_{n}, a_{n}, ξ_{n}), n = 0, 1, ...$ , with state-action dependent discount factors of the form $α_{n} (x_{n}, a_{n})$ , where $a_{n}$ and $ξ_{n}$ are the control and the random disturbance at time $n$ , respectively. Assuming that the sequences of functions ${α_{n}}$ , ${c_{n}}$ and ${G_{n}}$ converge, in certain sense, to $α_{\infty}$ ,...

Optimality of the range for which equivalence between certain measures of smoothness holds

Z. Ditzian (2010)

Studia Mathematica

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Recently it was proved for 1 < p < ∞ that $ω^{m} {(f, t)}_{p}$ , a modulus of smoothness on the unit sphere, and $K ̃ ₘ {(f, t^{m})}_{p}$ , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence $ω^{m} {(f, t)}_{p} \approx K ̃ ₘ {(f, t^{r})}_{p}$ does not hold either for p = ∞ or for p = 1.