Abstract
This paper deals with a class of nonlinear optimization problems in a function space, where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type. Second-order necessary and sufficient optimality conditions are established, where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces. The results are applied to some optimal control problems for ordinary and partial differential equations.
Highlights
Let (X, S, μ) be a measure space with μ(X) < +∞
Whenever nonlinear optimal control problems are solved, second-order sufficient conditions play an essential role in the numerical analysis
We obtain necessary and/or sufficient conditions for local solutions (y, u) of (OC) by application of Theorems 2.1, 2.2, and 3.1 and Corollary 3.3, provided that the corresponding assumptions (2.1) and (A1)–(A3) are satisfied. We tacitly assume this in what follows and formulate these results in a way that is convenient for optimal control problems
Summary
Let (X, S, μ) be a measure space with μ(X) < +∞. In this paper we will study the following optimization problem:. Our main goal is to reduce the classical gap between the necessary and sufficient conditions for optimization problems in Banach spaces. In the case of finite dimensions, strongly active inequality constraints (i.e., with strictly positive Lagrange multipliers) are considered in the critical cone by associated linearized equality constraints. Speaking, this is what we are able to extend to infinite dimensions. (2.1) is equivalent to the independence of the derivatives {Gj(u)}j∈I0 in L∞(Xεu ) Under this assumption we can derive the first-order necessary conditions for optimality satisfied by u.
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