Differential inclusions the baire category method

1990 ◽  
pp. 104-136 ◽  
Author(s):  
Giulio Pianigiani
Keyword(s):  
Differential Inclusions ◽  
Baire Category

A few remarks on differential inclusions

2006 ◽  
Vol 136 (3) ◽  
pp. 649-668 ◽  
Author(s):  
M. A. Sychev
Keyword(s):  
Differential Inclusions ◽  
Simple Algorithm ◽  
Baire Category ◽  
Approximate Solutions ◽  
Specific Choice ◽  
Affine Functions ◽  
The Subject ◽  
Piecewise Affine Functions ◽  
Admissible Functions ◽  
Convergent Sequences

In this paper we analyse the methodology of the theory of differential inclusions. First, we emphasize that any sequence of piecewise affine functions with successive elements obtained by perturbations of preceding functions in the sets of their affinity converges strongly, together with the gradients. This gives a simple algorithm with which to construct sequences of approximate solutions that converge to exact solutions (neither the specific choice suggested by ‘the method of convex integration for Lipschitz functions’ nor Baire category methodology is required). We then suggest a functional that is defined in the set of admissible functions and measures maximal oscillations produced by sequences of admissible functions weakly convergent to a given function. This functional can be used to prove that the set of stable solutions is dense in the weak topology in the closure of the set of admissible functions either via the Baire category lemma or via a specific choice of strictly convergent sequences.We explain how the above-mentioned methods of finding solutions to differential inclusions are connected to earlier results on weak–strong convergence, i.e. to results on stability, in the calculus of variations and in differential inclusions.We also include information on developments in the subject in the three years after the results of this work were obtained.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and of order α ∈ (1, 2)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
JinRong Wang ◽  
Ahmed G. Ibrahim ◽  
Donal O’Regan ◽  
Adel A. Elmandouh
Keyword(s):  
Differential Inclusions ◽  
Mild Solutions ◽  
Multivalued Function ◽  
Linear Operators ◽  
Solution Set ◽  
Cosine Family ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Noninstantaneous Impulses

AbstractIn this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory.

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