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Second-order sensitivity relations and regularity of the value function for Mayer's problem in optimal control  [PDF]
Piermarco Cannarsa,Hélène Frankowska,Teresa Scarinci
Mathematics , 2014,
Abstract: This paper investigates the value function, $V$, of a Mayer optimal control problem with the state equation given by a differential inclusion. First, we obtain an invariance property for the proximal and Fr\'echet subdifferentials of $V$ along optimal trajectories. Then, we extend the analysis to the sub/superjets of $V$, obtaining new sensitivity relations of second order. By applying sensitivity analysis to exclude the presence of conjugate points, we deduce that the value function is twice differentiable along any optimal trajectory starting at a point at which $V$ is proximally subdifferentiable. We also provide sufficient conditions for the local $C^2$ regularity of $V$ on tubular neighborhoods of optimal trajectories.
The Mayer-Vietoris Property in Differential Cohomology  [PDF]
James Simons,Dennis Sullivan
Mathematics , 2010,
Abstract: In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differential cohomology functor J^ associated to any Z-graded cohomology functor J(, Z) which, in each degree, assigns to a point a finitely generated group. The approach is to show that the result follows from Diagram 1, the commutative diagram we take as a definition of differential cohomology, and Diagram 2, which combines the three Mayer-Vietoris sequences for J*(, Z), J*(, R) and J*(, R/Z).
The Mayer-Vietoris Property in Differential Cohomology  [PDF]
James Simons,Dennis Sullivan
Mathematics , 2010,
Abstract: In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differential cohomology functor J^ associated to any Z-graded cohomology functor J(,Z) which, in each degree, assigns to a point a finitely generated group. The approach is to show that the result follows from Diagram 1, the commutative diagram we take as a definition of differential cohomology, and Diagram 2, which combines the three Mayer-Vietoris sequences for J*(,Z), J*(,R) and J*(,R/Z).
Some optimal control problems for partial differential inclusions  [PDF]
Micha? Kisielewicz
Opuscula Mathematica , 2008,
Abstract: Partial differential inclusions are considered. In particular, basing on diffusions properties of weak solutions to stochastic differential inclusions, some existence theorems and some properties of solutions to partial differential inclusions are given.
Sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions  [PDF]
Ram U. Verma
International Journal of Stochastic Analysis , 2006, DOI: 10.1155/jamsa/2006/52041
Abstract: Some results on the sensitivity analysis for relaxed cocoercive quasivariational inclusions are obtained, which generalize similar sensitivity analysis results on strongly monotone quasivariational inclusions. Furthermore, some suitable examples of relaxed cocoercive mappings are illustrated.
Sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions
Ram U. Verma
International Journal of Stochastic Analysis , 2006,
Abstract: Some results on the sensitivity analysis for relaxed cocoercive quasivariational inclusions are obtained, which generalize similar sensitivity analysis results on strongly monotone quasivariational inclusions. Furthermore, some suitable examples of relaxed cocoercive mappings are illustrated.
Impulsive differential inclusions with constrains  [cached]
Tzanko Donchev
Electronic Journal of Differential Equations , 2006,
Abstract: In the paper, we study weak invariance of differential inclusions with non-fixed time impulses under compactness type assumptions. When the right-hand side is one sided Lipschitz an extension of the well known relaxation theorem is proved. In this case also necessary and sufficient condition for strong invariance of upper semi continuous systems are obtained. Some properties of the solution set of impulsive system (without constrains) in appropriate topology are investigated.
Full averaging of fuzzy impulsive differential inclusions
Natalia V. Skripnik
Surveys in Mathematics and its Applications , 2010,
Abstract: In this paper the substantiation of the method of full averaging for fuzzy impulsive differential inclusions is studied. We extend the similar results for impulsive differential inclusions with Hukuhara derivative (Skripnik, 2007), for fuzzy impulsive differential equations (Plotnikov and Skripnik, 2009), and for fuzzy differential inclusions (Skripnik, 2009).
Generalized flow invariance for differential inclusions  [PDF]
T. Gnana Bhaskar,V. Lakshmikantham
International Journal of Stochastic Analysis , 2006, DOI: 10.1155/jamsa/2006/49431
Abstract: We introduce a generalized notion of invariance for differential inclusions, using a proximal aiming condition in terms of proximal normals. A set of sufficient conditions for the weak and strong invariance in the generalized sense are presented.
Generalized flow invariance for differential inclusions
T. Gnana Bhaskar,V. Lakshmikantham
International Journal of Stochastic Analysis , 2006,
Abstract: We introduce a generalized notion of invariance for differential inclusions, using a proximal aiming condition in terms of proximal normals. A set of sufficient conditions for the weak and strong invariance in the generalized sense are presented.
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