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1.
N. U. Ahmed 《Journal of Optimization Theory and Applications》1985,45(1):147-157
In this note, we present the necessary conditions of optimality for time-optimal controls for a class of distributed-boundary control problems in general Banach spaces using the semigroup theory. Theorem 3.1 is based on a recent general maximum principle due to Barbu (Ref. 1), which was proved for strictly convex reflexive Banach spaces. Theorem 3.2 generalizes this result (for time-optimal control problems) by lifting the assumption.This work was supported by the National Science and Engineering Council of Canada under Grant No. 7109. 相似文献
2.
We first investigate strong convergence of the sequence generated by implicit and explicit viscosity approximation methods for a one-parameter nonexpansive semigroup in a real Banach space E which has a uniformly Gateaux differentiable norm and admits the duality mapping j ?? , where ?? is a gauge function on [0, ??). The main results also improve and extend some known results concerning the normalized duality mapping in the literature. 相似文献
3.
LetA be the infinitesimal generator of aC
0 semigroup in a Banach spaceE. We obtain necessary conditions for a solution of the Cauchy problem {fx112-1} to be classical for arbitrary ϕ εC([0,T]) andf εE. 相似文献
4.
Markus Kunze 《Semigroup Forum》2009,79(3):540-560
We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of
the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity.
We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on C
b
(E) which is continuous with respect to the strict topology β
0 is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β
0-continuous semigroups on C
b
(E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β
0-continuous. 相似文献
5.
G. V. Shevchenko 《Computational Mathematics and Mathematical Physics》2011,51(4):537-549
Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control
problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal
control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of
simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable,
it is proved that the minimizing sequence converges to an ɛ-optimal solution after a finite number of iterations. A pair {T, u(·)} is called an ɛ-optimal solution if |T − T
opt| − ɛ, where T
opt is the optimal time required for moving the system from the initial state to the origin and u is an admissible control that moves the system to an ɛ-neighborhood of the origin over the time T. 相似文献
6.
We present sufficient conditions on a Gaussian Mehler semigroup on a reflexive Banach space Eto be induced by a single positive symmetric operator Q \in
, and give a counterexample which shows that this representation theorem is false in every nonreflexive Banach space with a Schauder basis. We also show that the transition semigroup of a Gaussian Mehler semigroup on a separable Banach space Eacts in a pointwise continuous way, uniformly on compact subsets of E, in the space BUC(E) of bounded uniformly continuous real-valued funtions on E. The transition semigroup is shown to be strongly continuous on BUC(E) if and only if S(t) = Ifor all t 0 相似文献
7.
Ruey-Jen Jang 《Integral Equations and Operator Theory》2001,39(3):292-304
LetE be a Banach lattice with order continuous norm and {T(t)}
t0 be an eventually compactc
0-semigroup of positive operators onE with generator A. We investigate the structure of the geometric eigenspace of the generator belonging to the spectral bound when the semigroup is ideal reducible. It is shown that a basis of the eigenspace can be chosen to consist of element ofE with certain positivity structure. This is achieved by a decomposition of the underlying Banach latticeE into a direct sum of closed ideals which can be viewed as a generalization of the Frobenius normal form for nonnegative reducible matrices. 相似文献
8.
M. S. Nikol’skii 《Computational Mathematics and Modeling》2007,18(2):148-156
The article examines the convergence of the popular gradient projection method for optimal control problems with fixed time
and a free right-hand end point. General conditions are derived that substantiate weak convergence of the sequence of control
iterations to the set of extremum controls satisfying Pontryagin’s maximum principle. Under stronger assumptions we prove
strong convergence of the sequence in the Banach L
1-norm.
__________
Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 139–148, 2003. 相似文献
9.
H. O. Fattorini 《Applied Mathematics and Optimization》1974,1(2):163-188
We show here that a variant of Pontryagin's maximum principle holds for time-optimal trajectories of the systemu′ = Au + f that begin and end at the domain ofA. HereA is the infinitesimal generator of a strongly continuous semigroupS(·) in the reflexive Banach spaceE. This partly generalizes a result of Balakrishnan ([1]) where no conditions are assumed on the trajectories but whereS(t) is supposed to beonto for allt ≥ 0. A weakened version of the maximum principle is then proved forall time-optimal trajectories whenA generates an analytic semigroup andE is a Hilbert space. Finally, the results are modified to handle two examples involving the heat equation where the spaceE is not reflexive. 相似文献
10.
V. M. Veliov 《Journal of Optimization Theory and Applications》1987,54(3):541-563
Using the notion of the local convexity index, we characterize in a quantitative way the local convexity of a set in then-dimensional Euclidean space, defined by an integral of a multivalued mapping. We estimate the rate of convergence of the conditional gradient method for solving an abstract optimization problem by means of the convexity index of the constraining set at the solution point. These results are applied to the qualitative analysis of the solutions of time-optimal and Mayer problems for linear control systems, as well as for estimating the convergence rate of algorithms solving these problems. 相似文献
11.
Mild solution of semilinear impulsive integro‐differential evolution equation in Banach spaces
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Through solving the problem step by step and by applying the method of a C0 semigroup of operators combined with the Banach contraction theorem, we investigate the existence and uniqueness of a mild solution of semilinear impulsive integro‐differential evolution equation in Banach spaces. In addition, an explicit iterative approximation sequence of the mild solution is derived. The assumed conditions in the present theorems are weaker and more general, and the results obtained are the generalizations and improvements of some known results. Examples are also given to illustrate our main results. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
12.
LetA be anm-accretive operator in a Banach spaceE. Suppose thatA
−10 is not empty and that bothE andE
* are uniformly convex. We study a general condition onA that guarantees the strong convergence of the semigroup generated by—A and of related implicit and explicit iterative schemes to a zero ofA. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also
establish strong convergence under the assumption that the interior ofA
−10 is not empty. In Hilbert space this result is due to H. Brezis.
Sponsored by the United States Army under Contract No. DAAG29-75-C-0024. 相似文献
13.
《Quaestiones Mathematicae》2013,36(2):97-129
Synopsis (for ‘Evolution Problems involving non-stationary Operators between two Banach Spaces I-II) In this series of two papers the initial-value problem [B(t)u(t)' = A(t)u(t), Bu(0) = y, with A = A(t) and B = B(t) time-varying operators from one Banach space X to another Banach space Y, and y an arbitrary element of Y, is considered. By making use of the theory of B-evolutions and by integrating certain temporally inhomogeneous equations, a unique solution is obtained for any y in Y. The solution is formulated explicitly in terms of a certain solution operator which involves the B(t)-evolution generated by the closed pair >A(t),B(t)< of operators. Certain properties of the solution operator are also studied. The well-known results, obtained by making use of semigroup theory, for the evolution problem [u(t)]' = A(t)u(t), u(0) = u0, where A is a closed operator in a Banach space with dense domain, may also be derived from our results. 相似文献
14.
After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey
convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz
space (l
∞
,τ(l
∞
,l
1)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indE
a, a∈A, ofseparable Banach spacesE
a, and every Mackey null sequence inE is a null sequence in someE
a. IfE is ultrabornological, thenE can be represented as lim indE
a,a∈A, allE
a separable Banach spaces, such that every fast null sequence inE is a null sequence in someE
a. 相似文献
15.
Sander C. Hille 《Integral Equations and Operator Theory》2005,53(4):597-601
A linear semigroup in a Banach space induces a linear semigroup on a Banach space that can be continuously embedded in the
former such that its image is invariant. This restriction need not be strongly continuous, although the original semigroup
is strongly continuous. We show that norm or weak compactness of partial orbits is a necessary and sufficient condition for
strong continuity of the restriction of a C0-semigroup. We then show that if the embedded Banach space is reflexive and the norms of the restricted semigroup operators
are bounded near the initial time, then the restricted semigroup is strongly continuous. 相似文献
16.
The paper presents a closure theorem for the attainable trajectories of a class of control systems governed by a large class of nonlinear evolution equations in reflexive Banach spaces. Several existence theorems for optimal controls are proven that include a terminal control problem, a time-optimal control problem, and a special Bolza problem. Some results of independent interest are also presented.This work was supported in part by the National Research Council of Canada under Grant No. 7109.The authors would like to thank Professor L. Cesari for pointing out that joint continuity off is required for the setsG andR to satisfy the upper semicontinuity property (Theorems 5.1 and 5.2). 相似文献
17.
We give several applications of Rademacher sequences in abstract Banach lattices. We characterise those Banach lattices with an atomic dual in terms of weak* sequential convergence. We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the compactness of operators from Lp into Lq. Finally we generalise earlier work of ours by showing that, amongst Banach lattices F with an order continuous norm, those having the property that the linear span of the positive compact operators fromE into F is complete under the regular norm for all Banach lattices E are precisely the atomic lattices. 相似文献
18.
We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t)dt+dW
H
(t), t0, where A is the generator of a C
0-semigroup S on a separable real Banach space E and W
H
(t)
t0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup S
H
. 相似文献
19.
Valeria Marraffa 《Journal of Theoretical Probability》1988,1(3):255-261
It is shown here that for any Banach spaceE-valued amart (X
n) of classB, almost sure convergence off(Xn) tof(X) for eachf in a total subset ofE
* implies scalar convergence toX. 相似文献
20.
Charles Castaing 《Annali di Matematica Pura ed Applicata》1993,164(1):51-75
Summary New results of decomposition for bounded sequences in L
E
1
and in the space of integrably bounded multifunctions with non empty convex weakly compact values in a Banach space E and its applications to problems of Minimization, convergence of martingales, Multivalued Fatou lemma are presented. 相似文献