Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control of Semilinear Parabolic Equations
- PDF / 518,037 Bytes
- 25 Pages / 439.642 x 666.49 pts Page_size
- 58 Downloads / 216 Views
Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control of Semilinear Parabolic Equations Tuan Nguyen Dinh1 Received: 25 December 2019 / Accepted: 13 September 2020 / © Springer Nature B.V. 2020
Abstract We consider multiobjective optimal control problems for semilinear parabolic systems subject to pointwise state constraints, integral state-control constraints and pointwise state-control constraints. In addition, the data of the problems need not be twice Fr´echet differentiable. Employing the second-order directional derivative (in the sense of DemyanovPevnyi) for the involved functions, we establish necessary optimality conditions, via second-order Lagrange multiplier rules of Fritz-John type, for local weak Pareto solutions of the problems. Keywords Multiobjective optimal control · Semilinear parabolic equation · Necessary second-order optimality condition · Local weak Pareto solution · Second-order directional derivative Mathematics Subject Classification (2010) 35J25 · 49K20 · 49K27 · 90C29 · 90C46
1 Introduction The aim of our note is to explore necessary second-order optimality conditions for local optimal solutions, in the sense of Pareto, to a mutiobjective semilinear parabolic optimal control problem with a pure state constraint and mixed state-control constraints (denoted by (MOCP) in the next). The conditions of this kind refine first-order conditions with secondorder information which is much helpful to design numerical algorithms for computing optimal solutions. Let us mention that some variants of problem (MOCP) with the scalarvalued objective function (i.e. the single-objective problem) have attracted the attention of many researchers in various publications (see for example, [15, 17, 49, 50, 57] and some references given therein). This work was supported by a Grant of the UEH Foundation for Academic Research. Tuan Nguyen Dinh
[email protected] 1
Department of Mathematics and Statistics, and Institute of Applied Mathematics, University of Economics Ho Chi Minh City, 59C Nguyen Dinh Chieu, D.3, Ho Chi Minh City, Vietnam
T. Nguyen Dinh
Over the past years, a large number of articles has been devoted to the analysis of optimality conditions for single-objective optimal control models of partial differential equations (PDEs). In the case of optimal control settings with semilinear parabolic equations, we can list some obtained results such as first-order necessary optimality conditions via the method of Lagrange multipliers in [49, 57], first-order ones in the form of Pontryagin’s principles in [13, 14, 51, 52], and second-order necessary and/or sufficient conditions for optimality in [5, 10, 15, 17, 50, 54, 55, 57]. It is also noticed that the theory of optimality conditions are extensively developed for control problems governed by linear/semilinear elliptic systems in [4, 11, 34, 35, 56, 57, 59] and models driven by hyperbolic equations in [41, 42]. Recently, multiobjective optimal control frameworks with differential equations have been of great interest because of their im
Data Loading...
