Error estimates for discretized differential inclusions

Dontchev, Asen L.; Farkhi, Elza

doi:10.1007/bf02241223

Cited by 93 publications

(59 citation statements)

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“…Finite dimensional results analogous to ours can be found in [9,10,14] and [17]. Dontchev studied in [10] a differential inclusion without uniqueness for which he showed that there exists a discrete solution, approximating one of the many exact solutions with an order of precision larger than one; however, this result is non constructive since it does not tell us how to obtain the appropriate discrete solution.…”

Section: ∀(X Y) ∈ V × V Y ∈ ∂φ(X) ⇐⇒ ∀Z ∈ V φ(Z) − φ(X) ≥ Y Z − mentioning

confidence: 82%

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Numerical precision for differential inclusions with uniqueness

Bastien¹,

Schatzman

2002

ESAIM: M2AN

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Abstract.In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this results is the existence of solutions in cases which had not been previously treated. The order of these schemes is 1/2 in general and 1 when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl's rheological model, our estimates in maximum norm do not depend on spatial dimension.

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Section: ∀(X Y) ∈ V × V Y ∈ ∂φ(X) ⇐⇒ ∀Z ∈ V φ(Z) − φ(X) ≥ Y Z − mentioning

confidence: 82%

“…on ]0, T [, (1.8) 9) where B is a pseudo-monotone mapping from the Banach space V to its dual V , g is a function from L 2 (0, T ; V ) and ∂φ is the sub-differential of a convex proper and lower semi-continuous function φ from V to ] − ∞, +∞]; this sub-differential is defined by…”

Section: U(t) + B(u(t)) + ∂φ(U(t)) G(t)mentioning

confidence: 99%

Numerical precision for differential inclusions with uniqueness

Bastien¹,

Schatzman

2002

ESAIM: M2AN

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“…Following [3], we consider the so-called averaged modulus of continuity for the multifunction F (x, y, z, t) with…”

Section: X(t) ∈ G(x(t) X(t − ∆)ẋ(T −mentioning

confidence: 99%

Optimal control of delay systems with differential and algebraic dynamic constraints

Mordukhovich¹,

Wang

2005

ESAIM: COCV

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Abstract. This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between "slow" and "fast" variables. We pursue a twofold goal: to study variational stability for this class of control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems under consideration and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. The authors are not familiar with any results in these directions for such systems even in the delay-free case. In the first part of the paper we establish the value convergence of discrete approximations as well as the strong convergence of optimal arcs in the classical Sobolev space W 1,2 . Then using discrete approximations as a vehicle, we derive necessary optimality conditions for the initial continuous-time systems in both Euler-Lagrange and Hamiltonian forms via basic generalized differential constructions of variational analysis.Mathematics Subject Classification. 49J53, 49K24, 49K25, 49M25, 90C31, 93C30.

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“…Following [7], we consider the so-called averaged modulus of continuity for the multifunction F(x,y,t) with…”

Section: Discrete Approximations Of Neutral Inclusionsmentioning

confidence: 99%

Discrete approximations and necessary optimality conditions for functional-differential inclusions of neutral type

Mordukhovich

Wang

2004

2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)

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Abstract-This paper deals with necessary optimality conditions for optimal control systems governed by constrained functional-differential inclusions of neutral type. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of endpoint constraints. Developing the method of discrete approximations and employing advanced tools of generalized differentiation, we conduct a variational analysis of neutral functionaldifferential inclusions and obtain new necessary optimality conditions of both Euler-Lagrange and Hamiltonian types.

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Error estimates for discretized differential inclusions

Cited by 93 publications

References 1 publication

Numerical precision for differential inclusions with uniqueness

Numerical precision for differential inclusions with uniqueness

Optimal control of delay systems with differential and algebraic dynamic constraints

Discrete approximations and necessary optimality conditions for functional-differential inclusions of neutral type

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