Search ResultsShowing 1-9 of 9
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arXiv:1901.02588 (Published 2019-01-09)
Maximum principle for state constrained optimal control problems governed by multisolution p-Laplacian elliptic equations in the absence of convexity
Comments: 25 pagesCategories: math.OCA state constrained optimal control problem governed by a class of multisolution p-Laplacian elliptic equations is studied in this paper. Both the control domain and cost functional considered may be non-convex. Combining the multiplicity and degeneracy of the state equation with the non-convex assumptions is the main difficulty we will overcome. By transforming the initial problem to a well-posed and non degenerate problem with a point-point mixed constraint and then using Ekeland's variational principle, the Pontryagin's maximum principle for the initial problem is obtained by passing to the limits twice.
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arXiv:1810.00152 (Published 2018-09-29)
Optimization of the Principal Eigenvalue for Elliptic Operators
Comments: 44 pagesCategories: math.OCMaximization and minimization problems of the principle eigenvalue for divergence form elliptic operators with Dirichlet boundary condition are considered. The principal eigen map of elliptic operator is introduced and the continuity as well as the differentiability of such a map, with respect to the parameter in the diffusibility matrix, is established. For maximization problem, the admissible control set is convexified to get the existence of an optimal convexified relaxed solution. Whereas, for minimization problem, the relaxation of the problem under H-convergence is used to get an optimal $H$-relaxed solution. Some necessary optimality conditions are presented for both problems and illustrative examples are presented as well.
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arXiv:1703.08649 (Published 2017-03-25)
Second-Order Necessary Conditions for Optimal Control of Semilinear Elliptic Equations with Leading Term Containing Controls
Comments: 28 pagesCategories: math.OCAn optimal control problem for a semilinear elliptic equation of divergence form is considered. Both the leading term and the semilinear term of the state equation contain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.
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arXiv:1311.4402 (Published 2013-11-18)
Optimal Blowup Time for Controlled Ordinary Differential Equations
Comments: 23 pagesCategories: math.OCBoth the shortest and the longest blowup time for a controlled system are considered. Existence result and maximum principle for optimal triple are established. Thanks to some monotonicity of the controlled system, some kinds of "the front part local optimality" for optimal triple is established. Then proofs of the main results become easy, clear and abundant.
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arXiv:1309.5800 (Published 2013-09-23)
Time optimal control problems for some non-smooth systems
Comments: 23 pagesCategories: math.OCTime optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered.
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Necessary and Sufficient Conditions for Distinguishability of Linear Control Systems
Comments: 13 pagesDistinguishability takes a crucial rule in studying observability of hybrid system such as switched system. Recently, for two linear systems, Lou and Si gave a condition not only necessary but also sufficient to the distinguishability of linear systems. However, the condition is not easy enough to verify. This paper will give a new equivalent condition which is relatively easy to verify.
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arXiv:1012.0989 (Published 2010-12-05)
Cesari-type Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls
Comments: 24 pagesCategories: math.OCAn optimal control problem governed by semilinear elliptic partial differential equations is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.
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arXiv:1008.3360 (Published 2010-08-19)
Optimality Conditions for Semilinear Parabolic Equations with Controls in Leading Term
Comments: 22 apges, 1 figuresCategories: math.OCAn optimal control problem for semilinear parabolic partial differential equations is considered. The control variable appears in the leading term of the equation. Necessary conditions for optimal controls are established by the method of homogenizing spike variation. Results for problems with state constraints are also stated.
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arXiv:1008.1020 (Published 2010-08-05)
Second-Order Necessary/Sufficient Conditions for Optimal Control Problems in the Absence of Linear Structure
Categories: math.OCSecond-order necessary conditions for optimal control problems are considered, where the ``second-order" is in the sense of that Pontryagin's maximum principle is viewed as a first-order necessary optimality condition. A sufficient condition for a local minimizer is also given.