Optimal syntheses for Bolza Problems

Geometric Control Approach 65 A X2 X1 C +1 +1 C B -1 Figure 5: The synthesis for the control problem 22, 23. The sketched region is reached by curves that start from the origin with control −1 and then switch to +1 control between the points A and B. tion: ¨y = −e y + ˙y + 1. The associated control system is: ˙x 1 = x 2 22 ˙x 2 = −e x 1 + x 2 + 1 + u 23 We have ˙γ + 2 = 0 on the curve x 2 = e x 1 − 2. After meeting this curve, the γ + trajectory meets the x 1 -axis. Now the synthesis has a different shape because the trajectories corresponding to control −1 satisfy ˙γ 2 = 0 on the curve: x 2 = e x 1 24 that is contained in the half plane {x 1 , x 2 : x 2 }. Hence γ − never meets the curve given by 24 and this means that m = 0. Since we know that n is at least 1, by Remark 4, we have n = 1, m = 0. The synthesis is drawn in fig. 5.

5. Optimal syntheses for Bolza Problems

Quite easily we can adapt the previous program to obtain information about the optimal synthe- ses associated in the previous sense to second order differential equations, but for more general minimizing problems. We have the well known: L EMMA 7. Consider the control system: ˙x = Fx + uGx, x ∈ 2 , F, G ∈ 3 2 , 2 , F0 = 0, |u| ≤ 1 . 25 66 U. Boscain – B. Piccoli Let L : 2 → be a 3 bounded function such that there exists δ 0 satisfying Lx δ for any x ∈ 2 . Then, for every x ∈ 2 , the problem: min R τ Lxt dt s.t. x0 = 0, xτ = x , is equivalent to the minimum time problem with the same boundary conditions for the control system ˙x = FxLx + uGxLx. By this lemma it is clear that if we have a second order differential equation with a bounded- external force ¨y = f y, ˙y+u, f ∈ 3 2 , f 0, 0 = 0, |u| ≤ 1, then the problem of reaching a point in the configuration space y , v from the origin, minimizing R τ Lyt , ˙yt dt, under the hypotheses of Lemma 7 is equivalent to the minimum time problem for the system: ˙x 1 = x 2 Lx, ˙x 2 = f xLx + 1Lxu. By setting: α : 2 →]0, 1δ[, αx := 1Lx, β : 2 → , βx := f xLx, we have: Fx = x 2 α x, βx, Gx = 0, αx. From these it follows: 1 A x = x 2 α 2 , 1 B x = α 2 α + x 2 ∂ 2 α . The equations defining turnpikes are: 1 A 6= 0, 1 B = 0, that with our expressions become the differential condition α + x 2 ∂ 2 α = 0 that in terms of L is: Lx − x 2 ∂ 2 Lx = 0 26 R EMARK 4. Since L 0 it follows that the turnpikes never intersect the x 1 -axis. Since 26 depends on Lx and not on the control system, all the properties of the turnpikes depend only on the Lagrangian. Now we consider some particular cases of Lagrangians. L=Ly In this case the Lagrangian depends only on the position y and not on the velocity ˙y i.e. L = Lx 1 . 26 is never satisfied so there are no turnpikes. L=L ˙y In this case the Lagrangian depends only on velocity and the turnpikes are horizontal lines. L=V y + 1 2 ˙y 2 In this case we want to minimize an energy with a kinetic part 1 2 ˙y 2 and a pos- itive potential depending only on the position and satisfying V y 0. The equation for turnpikes is x 2 2 = 2V x 1 . References [1] A GRACHEV A., Methods of Control Theory, in “Mathematic Geometry”, Proc. ICM 94, Dekker, New York 1998. [2] A GRACHEV A., B ONNARD B., C HYBA M., K UPKA I., Sub Riemannian Sphere in Mar- tinet Flat Case, ESAIM: COCV 2 1997, 337–448. [3] A GRACHEV A., E L A LAOUI C., G AUTHIER J. P., Sub Riemannian Metrics on 3 , to ap- pear in Geometric Control and Non-Holonomic Problems in Mechanics, Conference Pro- ceeding Series, Canad. Math Soc. [4] A GRACHEV A., G AMKRELIDZE R., Simplectic Methods for Optimization and Control, in “Geometry of Feedback and Optimal Control”, edited by B. Jakubczyk and W. Respondek, Dekker, New York 1998. [5] B ARDI M., C APUZZO -D OLCETTA I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equation, Birkauser Boston, Inc., Boston MA 1997. Geometric Control Approach 67 [6] B IANCHINI R. M., S TEFANI G., Graded Approximations and Controllability along a Tra- jectory, SIAM J. Control and Optimization 28 1990, 903–924. [7] B OLTIANSKII , Sufficient Conditions for Optimality and the Justification of the Dynamic Programming Principle, SIAM J. Control and Optimization 4 1966, 326–361. [8] B RESSAN A., P ICCOLI B., A Baire Category Approach to the Bang-Bang Property, Jour- nal of Differential Equations 116 2 1995, 318–337. [9] B RESSAN A., P ICCOLI B., Structural Stability for Time-Optimal Planar Syntheses, Dy- namics of Continuous, Discrete and Impulsive Systems 3 1997, 335-371. [10] B RESSAN A., P ICCOLI B., A Generic Classification of Time Optimal Planar Stabilizing Feedbacks, SIAM J. Control and Optimization 36 1 1998, 12–32. [11] B RUNOVSKI P., Every Normal Linear System Has a Regular Time-Optimal Synthesis, Math. Slovaca 28 1978, 81–100. [12] B RUNOVSKY P., Existence of Regular Syntheses for General Problems, J. Diff. Equations 38 1980, 317–343. [13] F LEMING W. H., S ONER M., Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York 1993. [14] F ULLER A. T., Relay Control Systems Optimized for Various Performance Criteria, Auto- matic and remote control, Proc. first world congress IFAC Moscow 1, Butterworths 1961, 510–519. [15] J URDJEVIC V., Geometric Control Theory, Cambridge University Press 1997. [16] M ALISOFF M., On The Bellmann Equation for Control Problem with Exit Time and Un- bounded Cost Functionals, to appear on Proc. of 39th CDC of IEEE. [17] M ARCHAL

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