Limit solutions for control systems

Transcription

1 Limit solutions for control systems M. Soledad Aronna Escola de Matemática Aplicada, FGV-Rio Oktobermat XV, 19 e 20 de Outubro de 2017, PUC-Rio

2 1 Introduction & motivation 2 Limit solutions 3 Commutative case 4 Noncommutative case 5 Concluding remarks & References

3 Optimal Control, example 1: Fishing problem [Clark, 1976] State variable: x is the size of the fish population Control variable: u is the fishing effort Cost: net revenue in a fixed time interval T ( max Eu(t) c ) 0 x(t) u(t) dt, ẋ(t) = r x(t) (1 x(t)/k) u(t), 0 u(t) U max, a.e. on [0, T ], x(t) 0 on [0, T ], x(0) = x 0, x(t ) free. E : selling cost, c/x(t) : fishing cost, r : reproduction rate r/k : mortality due to space competition

4 Optimal Control, example 2: the Goddard problem in 1D (vertical motion of a rocket) [Goddard, 1919] [Seywald & Cliff, 1993] State variables: (h, v, m) are the position, speed and mass. Control variable: u is the normalized thrust (proportion of T max ) Cost: final mass of the rocket (or minimize fuel consumption) max m(t ), s.t. ḣ(t) = v(t), v(t) = 1/h(t) 2 + 1/m(t) ( T max u(t) D(h(t), v(t)) ), ṁ(t) = b T max u(t), 0 u(t) 1, a.e. on [0, T ], h(0) = 0, v(0) = 0, m(0) = 1, h(t ) = 1, T : free final time, b : fuel consumption coefficient, T max is the maximal thrust, D(h, v) is the drag

5 Standard Optimal control problem A standard optimal control problem is generally written as max T 0 L(t, x(t), u(t))dt + φ(t, x(t )), [cost] ẋ(t) = f(t, x(t), u(t)), p.c.t. t [0, T ], [dynamics] x(0) = x 0, [initial condition] (T, x(t )) S IR n+1, [final constraints] u(t) U IR m. [control constraint] (OCP) In general, u L 1 ([0, T ]; U) and x W 1,1 ([0, T ]; IR n ).

6 Example 3: Impulsive Control System [Bressan & Piccoli, 2007] Boy riding swing: changes the radius of oscillation θ(t) : angle, r(t) : radius of oscillation Kinetic and potential energies: T (r, θ, ṙ, θ) = m(ṙ2 + r 2 2 θ2 ), V (r, θ) = mgr cos θ. Set L := T V for the Lagrangian, then equation of motion for θ(t) : d dt L L = θ θ

7 Example 3: equations of motion We get 2mr θṙ + mr 2 θ = mgr sin θ. Setting ω := θ for the angular velocity, u(t) := r(t) for the control, θ = ω, ω = g sin θ u 2 ω u u. If we consider the problem of maximizing the angle max θ(t ) we find that the optimal control u(t) jumps at the position θ = 0!

8 Example 3: equations of motion We get 2mr θṙ + mr 2 θ = mgr sin θ. Setting ω := θ for the angular velocity, u(t) := r(t) for the control, θ = ω, ω = g sin θ u 2 ω u u. If we consider the problem of maximizing the angle max θ(t ) we find that the optimal control u(t) jumps at the position θ = 0!

9 AIM We aim at giving an appropriate interpretation of these impulsive control equations, for control functions u that may jump or even have unbounded variation!

10 The impulsive control system For a control equation of the form m ẋ = f(t, x, u, v) + g α (x) u α, α=1 for t [a, b], with x : [a, b] IR n being the state variable and u : [a, b] U IR m, v : [a, b] V IR l the control variables, we propose a notion of solution that is defined for L 1 inputs u (here v is a standard, bounded control), subsumes former concepts of solution.

11 The impulsive control system For a control equation of the form m ẋ = f(t, x, u, v) + g α (x) u α, α=1 for t [a, b], with x : [a, b] IR n being the state variable and u : [a, b] U IR m, v : [a, b] V IR l the control variables, we propose a notion of solution that is defined for L 1 inputs u (here v is a standard, bounded control), subsumes former concepts of solution.

12 Some remarks When u AC([a, b]; IR m ), the system is standard and the solution is interpreted in the sense of Carathéodory. Some reasons for looking at discontinuous trajectories: 1) lack of coercivity in the cost, 2) large change of value in a very short amount of time [Azimov & Bishop, New trends in astrodynamics and applications: Optimal trajectories for space guidance. Annals of the NY Academy of Sciences, 2005] [Catllá et al., On Spiking models for synaptic activity and impulsive differential equations, SIAM Review, 2005] [Gajardo, Ramirez & Rapaport, Minimal time sequential batch reactors with bounded and impulse controls, SIAM J. Control Opt., 2007]

13 Crucial features Some basic issues are nowadays consolidated. For instance: a notion of solution in distributional-theoretical sense is wrong, [Hájek, 1985]; [Miller, 2003] Lie brackets [g α, g β ] := g β g α g α g β matter.

14 Good notions of solution already existed in two cases. CASE I: The commutative case [g α, g β ] 0. One defines the solution via density of AC in L 1 (we adopt this definition here). It exploits the commutativity via the Flow-box theorem. References: [A. Bressan & F. Rampazzo, 1991] [A.V. Sarychev, 1991]

15 CASE II: The noncommutative case [g α, g β ] 0, and controls u of bounded variation. The concept of graph completion. References: [R.W. Rishel, 1965] [A. Bressan and F. Rampazzo, 1988] [G.N. Silva and R.B. Vinter, 1996] [F.L. Pereira and G.N. Silva, 2000]

16 1 Introduction & motivation 2 Limit solutions 3 Commutative case 4 Noncommutative case 5 Concluding remarks & References

17 Limit solutions - Definitions Consider the Cauchy problem m ẋ = f(t, x, u, v) + g α (x) u α, x(a) = x. α=1 for t [a, b], Given x IR n, u L 1 ([a, b]; U) and v L 1 ([a, b]; V ) : 1 x : [a, b] IR n is a LIMIT SOLUTION if, for every τ [a, b], there exists (u τ k ) AC([a, b]; U) such that, the corresponding Carathéodory solutions (x τ k ) are uniformly bounded and satisfy (x τ k, u τ k)(τ) (x, u)(τ) + (x τ k, u τ k) (x, u) 1 0, when k. 2 A limit solution x : [a, b] IR n is a SIMPLE limit solution if (u τ k ) can be chosen independently of τ. 3 A simple limit solution x : [a, b] IR n is a BV-SIMPLE limit solution if the approximating inputs u k have equibounded variation.

18 Outline of the results Consistency with standard (Carathéodory) solutions Some existence results Uniqueness in the commutative case Comparison with graph completion solutions

19 Example 1 [H.J. Sussmann & W. Liu, 1991] g 1 (x) := (1, 0, x 2 ), g 2 (x) := (0, 1, x 1 ), [g 1, g 2 ] = (0, 0, 2), ẋ = g 1 (x) u 1 + g 2 (x) u 2, t [0, 1], x(0) = (0, 0, 0). For u (0, 0), the Carathéodory solution is x C (0, 0, 0). Approximating u by one gets u k (t) = (k 1/2 (cos kt 1), k 1/2 sin kt), x k (t) = (k 1/2 (cos kt 1), k 1/2 sin kt, t + k 1 sin kt), u k (0, 0), x k x(t) := (0, 0, t), uniformly on [0, 1]. u AC, and x AC is a simple limit solution (but not BV simple!) Lack of uniqueness: the solution depends on (u k ) (we ll see this holds in general whenever [g α, g β ] 0)

20 Example 2 ẋ = x u, x(0) = 1, t [0, 1]. For every bounded u L 1 ([0, 1]; IR), the map x(t) = e u(t) u(a) is a limit solution. Actually, since m = 1, we will prove that x is the unique limit solution. Yet, x is not a simple limit solution in general, since for instance, if u := 1 Q [0,1] is the Dirichlet function, there is no way of pointwise approximating u with one sequence of absolutely continuous u k.

21 1 Introduction & motivation 2 Limit solutions 3 Commutative case 4 Noncommutative case 5 Concluding remarks & References

22 The commutative case, [g α, g β ] 0 Theorem 1 (Characterization): The limit solution is characterized by the solution of a change of coordinates (given by the Flow-box theorem) as follows. Change of coordinates: Extend the vector fields setting ( ) ( ) gα g α :=, f f :=. 0 Then [ g α, g β ] = 0. e α Let Π: IR n IR m IR n be the canonical projection on the first factor: Π(x, z) := x, and ϕ : IR n IR m IR n be defined by ( ϕ(x, z) := Π (x, z)e ( zm gm)... e ( z1 g1)) = Π ((x, z)e m α=1 zα gα ). Consider the diffeomorphism φ : IR n IR m IR n IR m given by φ(x, z) := (ϕ(x, z), z).

23 Commutative case: change of coordinates (continuation) Then G := Dφ g, F = Dφ f are such that G α = ( ) 0 =, F n ( ) ( ) φi F = f z α e j =. α x i,j=1 j x i 0 Hence the differential system after the change of variables φ reads ξ = F (t, ξ, z, v), ż = u. With the change of coordinates we can reduce the system to a non-impulsive one!

24 Theorem 2 (Continuity w.r.t. data): For every x IR n, u L 1, v L 1 there exists a unique limit solution of m ẋ = f(t, x, u, v) + g α (x) u α, x(a) = x, α=1 for t [a, b], that depends continuously on initial data and control: x 1 (t) x 2 (t) + x 1 x 2 1 ] M [ x 1 x 2 + u 1 (a) u 2 (a) + u 1 (t) u 2 (t) + u 1 u 2 1. and also w.r.t. v in the L 1 -norm. Corollary (Consistency): When u AC, the Carathéodory solution coincides with the unique limit solution.

25 Example 3 ẋ = g 1 (x) u 1 + g 2 (x) u 2, x 1 (0) = x 2 (0) = 1, with g 1 (x) := (x 1, x 2 ) t, g 2 (x) := (0, x 1 ) t, then [g 1, g 2 ] 0. Case I: u 1 (t) := u k,1 (t) := { [ [ 0 on 0, on [ 1 2, 1], u 2 (t) := u k,2 (t) := ( (k + 1) t k+1 { [ ] 0 on 0, on ] 1 2, 1]. 0 on [ 0, 1 2 ] 1 ) k+1 on ] k+1, ] on ] 1 2, 1] 0 on [ ] 0, 1 2 (k + 1) ( ) t 1 2 on ] 1 2, ] 1 k+1 1 on ] k+1, 1],

26 Example 3 - continuation x 1 (t) := { [ [ 1 on 0, 1 2 e on [ 1 2, 1], x 2 (t) := Case II: We invert the controls, ũ 1 := u 2, ũ 2 := u 1, 1 on [ 0, 1 2 [ e on t = 1 2 2e on ] 1 2, 1] (u 1, u 2 ) = (ũ 1, ũ 2 ) on [0, 1]\{1/2} ũ k,1 := u k,2, ũ k,2 := u k,1, { [ ] 1 on 0, 1 1 on [ [ 0, 1 2 x 1 (t) := e on ] 2 1 2, 1], x 2 (t) := 2 on t = 1 2 2e on ]. 1 2, 1]

27 Example 3 - continuation x 1 (t) := { [ [ 1 on 0, 1 2 e on [ 1 2, 1], x 2 (t) := Case II: We invert the controls, ũ 1 := u 2, ũ 2 := u 1, 1 on [ 0, 1 2 [ e on t = 1 2 2e on ] 1 2, 1] (u 1, u 2 ) = (ũ 1, ũ 2 ) on [0, 1]\{1/2} ũ k,1 := u k,2, ũ k,2 := u k,1, { [ ] 1 on 0, 1 1 on [ [ 0, 1 2 x 1 (t) := e on ] 2 1 2, 1], x 2 (t) := 2 on t = 1 2 2e on ]. 1 2, 1]

28 Solutions coincide almost everywhere In general: Proposition: If the system is commutative and u, û L 1 ([a, b]; U) coincide a.e. in [a, b] (this is, u = û in L 1 ) and verify u(a) = û(a), one has x(t) = ˆx(t)e m α=1 (uα(t) ûα(t))gα, t [a, b]. Hence, x(t) = ˆx(t), a.e. on [a, b].

29 Example 3 - With non-commutative vector fields! Set now g 2 (x) := (0, x 2 1) t, then [g 1, g 2 ] = (0, x 2 1) t, and x 1 (t) := { [ [ 1 on 0, 1 2 e on [ 1 2, 1], x 2 (t) := 1 on [ 0, 1 2 e e + e 2 on t = 1 2 on ] 1 2, 1] [ while x 1 (t) := { [ ] 1 on 0, 1 2 e on ] 1 2, 1], x 2 (t) := [ 1 on [ 0, e on t = 1 2 on ] 1 2, 1] So, in the general noncommutative case, the solutions do not coincide almost everywhere even if the control do.

30 1 Introduction & motivation 2 Limit solutions 3 Commutative case 4 Noncommutative case 5 Concluding remarks & References

31 The noncommutative case, u BV - Space-time system The existing notion of solution in the noncommutative case: For regular (absolutely continuous) controls u AC([a, b]; U), one can consider s: [a, b] [0, 1], s(t) := t + Var [a,t](u) b a + Var [a,b] (u), and reparametrize time ϕ 0 ( ) := s 1 ( ) with ϕ 0 : [0, 1] [a, b], and set ϕ(s) := u ϕ 0 (s), y := x ϕ 0. (ϕ 0, ϕ, y) is solution of the space-times system y 0(s) = ϕ 0(s), m y (s) = f(y 0 (s), y(s), ϕ(s), ψ(s))ϕ 0(s) + g α (y(s))ϕ α (s), (y 0, y)(0) = (a, x), α=1 s [0, 1].

32 u BV - Graph completions For BV controls u, let (ϕ 0, ϕ) be a graph completion of u : (ϕ 0, ϕ) : [0, 1] [a, b] U Lipschitz continuous such that, t [a, b], there exists s [0, 1] verifying (t, u(t)) = (ϕ 0, ϕ)(s).

33 Graph completion solutions Given u BV ([a, b]; U) and (ϕ 0, ϕ) : [0, 1] [a, b] U a graph completion of u, we let y be the solution of y 0(s) = ϕ 0(s), m y (s) = f(y 0 (s), y(s), ϕ(s), ψ(s))ϕ 0(s) + g α (y(s))ϕ α (s), (y 0, y)(0) = (a, x), α=1 s [0, 1]. Graph completion solution: (possibly) set-valued map x : [a, b] IR n, t = x(t) := y ϕ 1 0 (t). Single-valued graph completion solution: x : [a, b] IR n, σ : [a, b] [0, 1] a right-inverse of ϕ 0 such that (t, u(t)) = (ϕ 0, ϕ) σ(t), x(t) = y σ(t), for all t [a, b].

34 Representation of BV simple limit solutions Theorem 3 (Equivalence BV simple limit solution and single-valued graph completion solution): Given x IR n, u BV, v L 1 : x is a BV simple limit solution if and only if x is a single-valued graph completion solution. Ingredients of the proof: if : one has (t, u(t), x(t)) = (ϕ 0, ϕ, y) σ(t), with σ BV increasing. We prove pointwise approximation of σ with AC functions of uniform BV. only if : compactness (Ascoli-Arzelà theorem)

35 Existence of limit solutions for u BV An arcwise connected set U has the Whitney property is d(x, y) M x y, for some M > 0, where d is the geodesic distance. Theorem 4 (Existence of limit solution): If U has the Whitney property then, for any x IR n, u BV, v L 1, there exists a BV simple limit solution.

36 Consistency with Carathéodory solutions Theorem 5 (Consistency): If x IR n, u AC, v L 1, and x AC is a BV simple limit solution, then x is the Carathéodory solution. Recall Example 1!

37 1 Introduction & motivation 2 Limit solutions 3 Commutative case 4 Noncommutative case 5 Concluding remarks & References

38 Concluding remarks The same concept of solution is used both for the commutative and noncommutative cases. In the commutative case, the limit solutions are represented via a change of coordinates. In the noncommutative case, there is a representation of the BV simple limit solutions via the graph completion. When u BV : it is equivalent to approximate by AC functions with uniform BV, than to complete the graph.

39 References H.J. Sussmann. Ann. Probability, A. Bressan and F. Rampazzo. Boll. Un. Mat. Ital., A. Bressan and F. Rampazzo. J. Optim. Theory Appl., A.V. Sarychev. In Nonlinear synthesis (Sopron, 1989), volume 9 of Progr. Systems Control Theory, Birkhäuser Boston, Boston, MA, M.S. Aronna and F. Rampazzo. A note on systems with ordinary and impulsive controls. IMA J. Math. Control Inform., 2016 M.S. Aronna and F. Rampazzo. L 1 limit solutions for control systems. J. Differential Equations, 2015 M.S. Aronna, M. Motta and F. Rampazzo. Necessary conditions involving Lie brackets for impulsive optimal control problems, arxiv & in progress

40 OBRIGADA!