we are in case A or B. Hence there are no turnpikes; 2. the trajectory γ γ
0, we are in case A or B. Hence
nγ ≤ 3. If γ switches at 1 −1 A 0, by Lemma 2 all the others switchings of γ happen on the set 1 −1 A 0. Moreover, if γ switches to Y it has no more switchings. Hence nγ ≤ 3. The Theorem is proved with N x = 3. By direct computations it is easy to see that the generic conditions P 1 , . . . , P 9 , under which the construction of [19] holds, are satisfied under the condition: f x 1 , = ±1 ⇒ ∂ 1 f x 1 , 6= 0 14 that obviously implies f x 1 , = 1 or f x 1 , = −1 only in a finite number of points. In the framework of [9, 19, 18] we will prove that, for our problem 5, 6, with the condition 14, the “shape” of the optimal synthesis is that shown in fig. 3. In particular the partition of the reachable set is described by the following T HEOREM 2. The optimal synthesis of the control problem 5 6 with the condition 14, satisfies the following:1. there are no turnpikes; 2. the trajectory γ
± starting from the origin and corresponding to constant control ±1 exits the origin with tangent vector 0, ±1 and, for an interval of time of positive measure, lies in the set {x 1 , x 2 : x 1 , x 2 ≥ 0} respectively {x 1 , x 2 : x 1 , x 2 ≤ 0};3. γ
± is optimal up to the first intersection if it exists with the x 1 -axis. At the point in which γ + intersects the x 1 -axis it generates a switching curve that lies in the half plane {x 1 , x 2 : x 2 ≥ 0} and ends at the next intersection with the x 1 -axis if it exists. At that Geometric Control Approach 61 -1 X2 X1 C C +1 +1 +1 -1 -1 Figure 3: The shape of the optimal synthesis for our problem. point another switching curve generates. The same happens for γ − and the half plane {x 1 , x 2 : x 2 ≤ 0};4. let y
Parts
» Directory UMM :Journals:Journal_of_mathematics:OTHER:
» Introduction Agrachev COMPACTNESS FOR SUB-RIEMANNIAN
» Geodesics Agrachev COMPACTNESS FOR SUB-RIEMANNIAN
» Compactness Agrachev COMPACTNESS FOR SUB-RIEMANNIAN
» Second Variation Agrachev COMPACTNESS FOR SUB-RIEMANNIAN
» Existence of solutions by the PWB method
» Consistency properties and examples
» Continuous dependence under local uniform convergence of the operator
» Continuous dependence under increasing approximation of the domain
» Maximizing the mean escape time of a degenerate diffusion process
» Introduction Notations and preliminary results To each family f
» An admissible weight for the process 1 is a set of positive numbers, l Let l
» Explicit optimality conditions for the single input case
» If there exists an admissible weight l such that each bad bracket of l-weight
» Introduction Basic definitions Boscain – B. Piccoli GEOMETRIC CONTROL APPROACH
» Optimal synthesis Boscain – B. Piccoli GEOMETRIC CONTROL APPROACH
» Applications to second order equations
» we are in case A or B. Hence there are no turnpikes; 2. the trajectory γ γ
» By definition a turnpike is a subset of 1 We leave the proof to the reader. 3. Let γ
» The two assertions can be proved separately. Let us demonstrate only the first, being the proof
» Optimal syntheses for Bolza Problems
» Introduction Brandi – A. Salvadori ON MEASURE DIFFERENTIAL INCLUSIONS
» On asymptotic cone On property Q
» On measure differential inclusions, weak and strong formulations
» The main closure theorem Further closure theorems for measure differential inclusions
» On comparison with Stewart’s assumptions
» Sulle funzioni a variazione limitata, Ann. Scuola Nor. Sup. Pisa 5 1936,
» Introduction Bressan SINGULARITIES OF STABILIZING FEEDBACKS
» Nonexistence of continous stabilizing feedbacks
» Generalized solutions of a discontinuous O.D.E.
» Patchy vector fields Bressan SINGULARITIES OF STABILIZING FEEDBACKS
» Directionally continuous vector fields Stabilizing feedback controls
» Some open problems Bressan SINGULARITIES OF STABILIZING FEEDBACKS
» Notations, basic assumptions and abstract classes of dynamics
» Necessary conditions Bucci THE NON-STANDARD LQR PROBLEM
» Sufficient conditions Bucci THE NON-STANDARD LQR PROBLEM
» Introduction Ceragioli EXTERNAL STABILIZATION OF DISCONTINUOS SYSTEMS
» UBIBS Stability Ceragioli EXTERNAL STABILIZATION OF DISCONTINUOS SYSTEMS
» The Main Result Ceragioli EXTERNAL STABILIZATION OF DISCONTINUOS SYSTEMS
» Proof of Theorem 2 Ceragioli EXTERNAL STABILIZATION OF DISCONTINUOS SYSTEMS
» The augmented system Pandolfi
» “Parabolic” case: from the Frequency domain condition to the DI
» Introduction Rampazzo – C. Sartori ON PERTURBATIONS OF MINIMUM PROBLEMS
» A convergence result Rampazzo – C. Sartori ON PERTURBATIONS OF MINIMUM PROBLEMS
» Reparameterizations and Bellman equations i The functions f
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