Convexity of Values of Vector Integrals, Theorems on Measurable Choice and Variational Problems

Аркин, В. И.; Levin, V. L.

doi:10.1070/rm1972v027n03abeh001378

Cited by 37 publications

(22 citation statements)

References 51 publications

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“…In Section 3, we apply the result of Section 2 to the continuous-time allocation process previously described. In contrast with previous works in this direction [2,3,7,8,17], we admit the existence of some economic parameters in the consumption program besides the time and the novel constraint given by (1.2). For a discussion of the problem described in the previous papers but depending on a parameter we refer [5], where some limit behaviour when the parameter tends to a limit is also examined.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 71%

“…A one-dimensional version of this Liapunov-type result is proved in Lemma 2.1. It is optimal in the sense discussed in Remark 2.2 below and is close to the one given in Theorem 2.1 of [2] except for the "unilateral" condition (see (ii) in Lemma 2.1) and extends Lemma 2.2 in [1]. In Section 3, we apply the result of Section 2 to the continuous-time allocation process previously described.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 73%

“…Related problems can be found in [3,4,7,17] but in all of these papers the authors do not consider any other parameter (say x) besides the time and a constraint of the form (1.2). Even if the authors in [2] take others parameters into account, a constraint of the type (1.2) is absent. In fact the novelty of our formulation lies in (1.2).…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 91%

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A Variational Problem Related to a Continuous-Time Allocation Process for a Continuum of Traders

Flores-Bazán¹,

Raymond²

2001

Journal of Mathematical Analysis and Applications

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Section: Introduction and Formulation Of The Problemmentioning

confidence: 71%

Section: Introduction and Formulation Of The Problemmentioning

confidence: 73%

Section: Introduction and Formulation Of The Problemmentioning

confidence: 91%

See 1 more Smart Citation

A Variational Problem Related to a Continuous-Time Allocation Process for a Continuum of Traders

Flores-Bazán¹,

Raymond²

2001

Journal of Mathematical Analysis and Applications

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“…For u(t, z), we take some point where the maximum is attained, so that the function u(t, x) is Borel measurable. Such a function exists by virtue of the theorem of measurable choice [6].…”

Section: Let Us Define the Functions L(t = Re(t W(t Z) = G -1 (T)dmentioning

confidence: 99%

“…The strategies (17)-(19) are optimal guaranteed strategies for the problem (6)- (10): u(.) = u~ w(.)…”

Section: Let Us Define the Functions L(t = Re(t W(t Z) = G -1 (T)dmentioning

confidence: 99%

Optimal guaranteed control of linear systems under disturbances

Ivanov

1996

Math Notes

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ABSTRACT. This paper is devoted to the problem of the minimax control of a dynamical system with quadratic performance functional under external disturbances and geometric control constraints. The optimal guaranteed control strategy is obtained in explicit form.KEY WORDS: optimal control under disturbances, minimax control, optimal guaranteed control, geometric control constraints, performance functional.w Introduction Consider the linear controllable systemon the given time interval t 6 [to, T], where z 6 R" is the phase vector of the system, A(t), B(t), C(t) axe matrices of dimensions n • n, n • m, n x l, respectively, u 6 R m is the control, and v 6 R l is a disturbance. We shall study the optimal guaranteed control of the system, i.e., the control that minimizes a functional Y(z(-), u(-), v(. )) under the assumption that the disturbance v is most unfavorable. This problem can be treated within the scope of differential game theory [1, 2] developed by L. S. Pontryagin, N. N. Krasovskii, and by their disciples. In this approach, the control u(t) is treated as belonging to the first player who aims to minimize the functional J, and the disturbance v(t) is treated as the control of the second player who aims to maximize the same functional. In particular, as a physical example of this problem, we can consider the problem of the soft landing of an aircraft under wind disturbances [3].Let us recall the standard approach to the solution of such a problem [2, pp. 156-162]. For the system (1), we consider the quadratic performance functional f; J --+ vr(t)H(t)v(O) dr,where R, G (O , H(t) are quadratic symmetric positive semidefinite matrices of appropriate dimensions.We seek the minimax of this functional over geometrically unconstrained position strategies u (t, x(t)) and v (t, x(t)). It is known that the optimal control strategy is of the formwhere K(t) is an easily calculated matrix.However, this approach has a serious drawback. The fact is that in actual systems, as a rule, the control is geometrically constrained:This type of constraint cannot be replaced by the insertion of a quadratic penalty function into the performance functional J. In particular, for any choice of the functional (2), which determines the matrix K(t) (3), there exist large values of the phase vector x(t) such that the optimal control (3) falls outside the limits of the constraint (4).

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Some Results on Density of Extreme Selections for Measurable Multifunctions

Chûǒng

1986

Math. Nachr.

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The density property of the set of extreme selections considered in general ORLIOZ spaces ia established in topologies finer than the weak one for measurable multifunctions taking values in locally convex S W S L~ spaces and not necessarily integrable.Math. Na~hr. 126 (1986) from 9 into E is said (A, cW(E))-measurable, or simply, measurable if its graph, i.e. the Graph r = ((0,s) E 9 x E I 2 E r ( w ) } , belongs to d @ cW(E). For a function f : 9 + E such that f -( U ) E d for every open set U in E, we shall say that it is measurable in the weak sense. An arbitrary measure space (9, A, p ) with p 2 0 is said to be c q l e t e if for any N E d with p ( N ) = 0, one haa M E d for all M c N .Let us recall some facts which will be used in the paper. A) Let (9, d, p ) be a complete masure space with p 2 0, finite. For a function f : 9 + E, the following 8tate7llents are equivalent;pointwise limit of an ordinary 8equence of measurable functi.ne a.88thmin.g d) f is nwawrable in the usual 8ense. e) The function w H (f(w), g(w)) is maszlrable (in the uaual sense) for every ~ealarly nzeasurable function g: 9 + E'. Proof. The statements a) b) H c) # d) can befound in[7] (TheoremIII. 36) while the implication b) =+ e) follows immediately from c) + e) and the inverse one is B) With (Q, d , p ) given a8 above, E = n EL is the CABTESiUn product of locally m v e x s~ El, Ea, ..., Ek. 8uppme tha.4 for each 8, either E6 is Lusm or Ed' a8 well ae E6 ia Snsm. Then foreachscalarlymasurabb functiOng = (#):=,: 9 + 8' the mltifunctiOn V defined by: V ( w ) = {z = (a?):-l E E I max l(9, gL(w)),l < 1) is m m r a b l e , where (., denotes the canonical bilinear form between EL and (EL)'.Proof. It suffices to show that for each 8, the function (0,9) H (a?, #(w)), is d @ cfs(E6)-meaaurable. Indeed, since, clearly, the function (o,z) H (0, a?) is mmurable from 9 X E into 9 x Ea, the function (w, x) H (a?, #(w)), is d @ cfs(E)-measurable, hence so is the function (w, z) H Max I(*, @ (~) )~l .Consequently, Graph V E d @ cW(E).

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Convexity of Values of Vector Integrals, Theorems on Measurable Choice and Variational Problems

Cited by 37 publications

References 51 publications

A Variational Problem Related to a Continuous-Time Allocation Process for a Continuum of Traders

A Variational Problem Related to a Continuous-Time Allocation Process for a Continuum of Traders

Optimal guaranteed control of linear systems under disturbances

Some Results on Density of Extreme Selections for Measurable Multifunctions

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