Necessary and sufficient optimality conditions for a system described by a nonlinear elliptic equation

Serovaiskii, S. Ya.

doi:10.1007/bf00970485

Cited by 2 publications

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“…The established results, representing a development of previously reported works [8,9] by the present author give promise that extended derivatives may successfully replace the classical derivatives in various applications. Among such applications are numerous problems that arise in nonlinear analysis (the implicit function theorem and the Lusternik theorem), in differential topology (notions of differentiable manifold, tangent space, and tangent fibration), in the Lee group theory (relations between Lee groups and Lee algebras), in the theory…”

mentioning

confidence: 71%

“…If condition (5) is violated, then relation (8) no longer yields an a priori estimate for the solution of the linearized equation in the space Y , and the right-hand side of (8) even may be meaningless at all because of low regularity of v. Thus, in proving differentiability of the solution of the equation under examination with respect to the first term, one cannot use the inverse function theorem. With the latter fact, lack of differentiability of the operator under consideration still remains to be seen.…”

Section: Regular Casementioning

confidence: 94%

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Calculation of functional gradients and extended differentiation of operators

Serovajsky¹

2005

Journal of Inverse and Ill-posed Problems

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Qualitative and quantitative analysis of optimal-control problems and inverse problems in mathematical physics largely implies the calculation of functional gradients giving optimality criteria or closing errors. As a rule, this procedure is used in calculating necessary extremum conditions and in gradient methods. If a system under study is linear, then calculation of a functional gradient presents no serious difficulties. Yet, in nonlinear problems of mathematical physics there arise substantial difficulties stemming from the fact that, as the control (parameter to be identified) is put to variation, the state function of the system also undergo changes. Thus, for the functional gradient to be found, differentiability of the state function with respect to control must be established. Substantiation of this property for nonlinear infinite-dimensional systems is a point far from being trivial. Moreover, the example given below shows that, although the system of interest contains no nonsmooth expressions, the required property can be violated. This circumstance does not allow one to calculate functional gradients by standard methods and use traditional optimization methods. Nonetheless, the system under study displays a property that can be considered as a weak form of differentiability of the state function with respect to control. As a result, we arrive at the notion of extended derivative of an operator, presenting generalization of the classical types of operator derivatives. With the help of this notion, one can find a functional gradient even if its direct determination is impossible. In this way, we can enlarge the area of applicability of optimization methods. It is shown that, for extended derivatives, an analogue to the inverse function theorem can be formulated which makes it possible to give a general scheme permitting calculation of functional gradients in nonlinear problems of mathematical physics even in those cases in which the state function turns out to be a nondifferentiable function of the control variable.

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mentioning

confidence: 71%

Section: Regular Casementioning

confidence: 94%