# Construction of Feasible Trajectories When the Classical Inward Pointing Condition Is Violated

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## Nonsmooth First Order Necessary Optimality Conditions

Since optimal control problems deal with the problem of finding a control lawfor a given system such that a certain optimality criterion (or ‘objective’) is achieved, many mathematicians were interested in constructing conditions for which an optimal control can be derived. The study of such conditions goes back to the 1950s with the work of L. Pontryagin and his famous Pontryagin’s Maximum Principle, which provides a set of necessary conditions which an optimum (if it exists) must satisfy; and the Dynamic Programming Principle, which simplifies the search of an optimal control function to the task of finding the solution to a partial differential equation known as the Hamilton-Jacobi-Bellman equation: this procedure gives a sufficient condition for an optimum. This thesis project treats only necessary optimality conditions in the form of the Maximum Principle for control systems (ordinary differential equation) and in the form of the Extended Euler-Lagrange condition (and the associated optimality conditions) for the differential inclusion dynamics.

### Preliminary Result: Convergence of Measures

We consider closed subsets D and Di , for i = 1, 2, . . . of [S, T]⇥RK. We denote by D(.), Di (.) : [S, T] { RK the multifunctions defined as D(t) := {z 2 RK : (t, z) 2 D} and Di (t) := {z 2 RK : (t, z) 2 Di } for all i = 1, 2, . . . . Let {μi } be a convergent sequence of positive finite measures. Our aim now is to justify the limit-taking of sequences like d⌘i (t) = /i (t)dμi (t) i = 1, 2, . . . in which {/i (t)} is a sequence of Borel measurable functions satisfying /i (t) 2 Di (t) μi − a.e.

#### Nonsmooth State Constraint

The next lemma establishes the construction, locally in time, of a neighboring feasible trajectory verifying a W1,1−estimate in the case where the initial data x0 belongs to a ‘corner’ of the state constraint (i.e. x0 2 A\A0). Here, for the local construction, the existence of a vector belonging to the set of velocities and pointing inside the state constraint (i.e. assumption (A.3)) is not relevant.
Lemma 2.2.3. (cf. ) Fix any r0 > 0. Consider a multifunction F : Rn { Rn and a nonempty set A ⇢ Rn. Suppose that only hypotheses (A.1) and (A.2) are satisfied (for some constants c > 0, kF > 0 and R0 = ec(T−S) (r0 + 2)). Take any x0 2 (A \ A0) \ r0B, and any w0 2 int TA(x0). Then we can find ✓1 2 (0, 1 16 ), ↵ % 0, ⌧1 2 (0, T − S] and K1 > 0 such that the following property is satisfied: given any F-trajectory ˆ x(.) on [S,T] and  » % 0 such that 8>><>>: max t2[S,T] dA( ˆ x(t))   » ˆ x(S) 2 A \ (x0 + ✓1B) .

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Global Construction of Neighboring Feasible Trajectories with W1,1−Linear Estimates

This section is an extension of the local construction of neighboring feasible trajectories established in Section 2.2. The F−trajectory previously constructed has all the necessary properties of a feasible F−trajectory with one exception: it satisfies the state constraint only on a suitably small initial interval [S, S +⌧] (for some ⌧ 2 (0, T − S]). The idea behind the proof of extending the result to the whole time interval [S, T] is to proceed with a recursive construction and to construct a finite sequence of sub-arcs, whose concatenation is an F−trajectory satisfying the state constraint on the whole time interval [S, T] and which, at the same time, satisfies a linear, W1,1− estimate, with an increased constant of proportionality. By contrast, this construction requires a stronger assumption on the state constraint set. We shall therefore replace assumption (A.1) with (A.10) The set A ⇢ Rn is closed, non-empty and convex.

Notation
1 Preliminaries
1.1 Nonsmooth Analysis Tools
1.2 Optimal Control Problems
1.2.1 Control Systems and Differential Inclusions
1.2.2 Existence of Solutions and Minimizers
1.2.3 Nonsmooth First Order Necessary Optimality Conditions
1.3 State Constrained Optimal Control Problems
1.3.1 Description
1.3.2 Preliminary Result: Convergence of Measures
1.3.3 Nonsmooth Maximum Principle
1.3.4 Extended Euler-Lagrange Condition
2 Neighboring Feasible Trajectories and W1,1−Linear Estimates
2.1 Overview
2.2 Local Construction of Neighboring Feasible Trajectories with W1,1−Linear Estimates
2.2.1 Smooth State Constraint
2.2.2 Nonsmooth State Constraint
2.3 Global Construction of Neighboring Feasible Trajectories with W1,1−Linear Estimates
2.4 Proof of the Results
3 Non-Degenerate Forms of the Generalized Euler-Lagrange Condition
3.1 Motivation
3.1.1 Degeneracy Phenomenon
3.1.2 Constraint Qualification
3.2 Main Result
3.3 Example
3.4 Proof of the Main Result
4 Normality of the Generalized Euler-Lagrange Condition: Differential Inclusions
4.1 Introduction
4.2 Main Result
4.3 Example
4.4 Proof of the Main Result
5 Calculus of Variations Problems: Applications to Normality
5.1 Introduction
5.2 Main Results
5.3 Two Proof Techniques for the Main Results
5.3.1 Proof of Theorem 5.2.2
5.3.2 Proof of Theorem 5.2.3
5.4 Proofs of Proposition 5.3.4 and Lemma 5.3.2
6 Construction of Feasible Trajectories When the Classical Inward Pointing Condition Is Violated
6.1 Motivation
6.2 A Local Viability Result
6.3 Brockett Nonholonomic Integrator with Flat Constraint: Local Construction .
6.4 Neighboring Feasible Trajectories with Nonlinear W1,1−estimate
6.4.1 Conjecture
6.4.2 Examples: Brockett Nonholonomic Integrator
7 Necessary Optimality Conditions For Average Cost Minimization Problems
7.1 Introduction
7.2 Average on Measures with Finite Support
7.3 Main Results
7.4 Preliminary Results in Measure Theory
7.5 Proofs of Theorem 7.3.1 and Theorem 7.3.3
Appendix
A Continuity Sets
B Measurability of F(t, !)
Conclusions and Perspectives
Dissemination
Bibliography

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