Semiconcavity and optimal control: an intrinsic approach

Transcription

1 Semiconcavity and optimal control: an intrinsic approach Peter R. Wolenski joint work with Piermarco Cannarsa and Francesco Marino Louisiana State University SADCO summer school, London September 5-9, 2011 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

2 Outline 1 Optimal Control problems Optimal control Value functions and semiconcavity Differential Inclusions (DI) Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

3 Outline 1 Optimal Control problems Optimal control Value functions and semiconcavity Differential Inclusions (DI) 2 Smooth parameterizations? Necessary conditions Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

4 Outline 1 Optimal Control problems Optimal control Value functions and semiconcavity Differential Inclusions (DI) 2 Smooth parameterizations? Necessary conditions 3 New (DI) assumptions Examples Consequences Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

5 Outline 1 Optimal Control problems Optimal control Value functions and semiconcavity Differential Inclusions (DI) 2 Smooth parameterizations? Necessary conditions 3 New (DI) assumptions Examples Consequences 4 New idea and results A replacement for a priori estimates Semiconcavity results with (DI) Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

6 Optimal control problems Control Dynamics: (CD) ẋ(s) = f ( x(s), u(s) ) a.e. s [t, T ] u(s) U a.e. s [t, T ] x(t) = x, where f : R n R n R n is continuous in (x, u) and Lipschitz in x, the admissible control set U R m is compact, and u : [t, T ] R m is measurable. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

7 Optimal control problems Control Dynamics: (CD) ẋ(s) = f ( x(s), u(s) ) a.e. s [t, T ] u(s) U a.e. s [t, T ] x(t) = x, where f : R n R n R n is continuous in (x, u) and Lipschitz in x, the admissible control set U R m is compact, and u : [t, T ] R m is measurable. Two classic problems: 1. MinTime: Given a closed target set S R n, the problem is min (T t) over ( x( ), u( ) ) satisfying (CD) and x(t ) S. The optimal value T (x) is the minimum time function. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

8 Optimal control problems Control Dynamics: (CD) ẋ(s) = f ( x(s), u(s) ) a.e. s [t, T ] u(s) U a.e. s [t, T ] x(t) = x, where f : R n R n R n is continuous in (x, u) and Lipschitz in x, the admissible control set U R m is compact, and u : [t, T ] R m is measurable. Two classic problems: 1. MinTime: Given a closed target set S R n, the problem is min (T t) over ( x( ), u( ) ) satisfying (CD) and x(t ) S. The optimal value T (x) is the minimum time function. 2. Mayer problem: Given endpoint cost l : R n R, the problem is min l ( x(t ) ) over ( x( ), u( ) ) satisfying (CD). The optimal value V (t, x) is the value function. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

9 Regularity of value functions - SemiConCavity (SCC) A natural regularity property for T ( ) and V (, ) is the property of being semiconcave. A locally Lipschitz function g : R m R is semiconcave provided there exists k > 0 so that 1[ ] g(x + z) + g(x z) g(x) k z 2 2 x, z R n. (the three point property) Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

10 Regularity of value functions - SemiConCavity (SCC) A natural regularity property for T ( ) and V (, ) is the property of being semiconcave. A locally Lipschitz function g : R m R is semiconcave provided there exists k > 0 so that 1[ ] g(x + z) + g(x z) g(x) k z 2 2 x, z R n. (the three point property) A semiconcave function

11 Regularity of value functions - SemiConCavity (SCC) A natural regularity property for T ( ) and V (, ) is the property of being semiconcave. A locally Lipschitz function g : R m R is semiconcave provided there exists k > 0 so that 1[ ] g(x + z) + g(x z) g(x) k z 2 2 x, z R n. (the three point property) A semiconcave function x z x x + z

12 Regularity of value functions - SemiConCavity (SCC) A natural regularity property for T ( ) and V (, ) is the property of being semiconcave. A locally Lipschitz function g : R m R is semiconcave provided there exists k > 0 so that 1[ ] g(x + z) + g(x z) g(x) k z 2 2 x, z R n. (the three point property) A semiconcave function LHS x z x x + z

13 Regularity of value functions - SemiConCavity (SCC) A natural regularity property for T ( ) and V (, ) is the property of being semiconcave. A locally Lipschitz function g : R m R is semiconcave provided there exists k > 0 so that 1[ ] g(x + z) + g(x z) g(x) k z 2 2 x, z R n. (the three point property) A semiconcave function LHS x z x x + z Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

14 A geometric description of SCC A Lipschitz function g : R n R is (SCC) if and only if σ > 0 with g(x) = inf { q(x) : q(x) = σx 2 + bx + c, g(x) q(x) } = inf { q(x, α) : α A }, where (x, α) q(x, α) is C 1+ in x and continuous in (x, α). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

15 A geometric description of SCC A Lipschitz function g : R n R is (SCC) if and only if σ > 0 with g(x) = inf { q(x) : q(x) = σx 2 + bx + c, g(x) q(x) } = inf { q(x, α) : α A }, where (x, α) q(x, α) is C 1+ in x and continuous in (x, α).

16 A geometric description of SCC A Lipschitz function g : R n R is (SCC) if and only if σ > 0 with g(x) = inf { q(x) : q(x) = σx 2 + bx + c, g(x) q(x) } = inf { q(x, α) : α A }, where (x, α) q(x, α) is C 1+ in x and continuous in (x, α).

17 A geometric description of SCC A Lipschitz function g : R n R is (SCC) if and only if σ > 0 with g(x) = inf { q(x) : q(x) = σx 2 + bx + c, g(x) q(x) } = inf { q(x, α) : α A }, where (x, α) q(x, α) is C 1+ in x and continuous in (x, α).

18 A geometric description of SCC A Lipschitz function g : R n R is (SCC) if and only if σ > 0 with g(x) = inf { q(x) : q(x) = σx 2 + bx + c, g(x) q(x) } = inf { q(x, α) : α A }, where (x, α) q(x, α) is C 1+ in x and continuous in (x, α). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

19 Previous results yielding (SCC) There is considerable literature on value functions being (SCC). Most relevant here: Cannarsa, Frankowska, Sinestrari, McEneaney. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

20 Previous results yielding (SCC) There is considerable literature on value functions being (SCC). Most relevant here: Cannarsa, Frankowska, Sinestrari, McEneaney. Basic idea with (CD): (Illustration with Min Time) We seek an upper bound (by k z 2 ) of T (x + z) + T (x z) 2T (x). Take an optimal solution starting from x and use it to construct feasible solutions from x ± z that will yield the appropriate estimates. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

21 Previous results yielding (SCC) There is considerable literature on value functions being (SCC). Most relevant here: Cannarsa, Frankowska, Sinestrari, McEneaney. Basic idea with (CD): (Illustration with Min Time) We seek an upper bound (by k z 2 ) of T (x + z) + T (x z) 2T (x). Take an optimal solution starting from x and use it to construct feasible solutions from x ± z that will yield the appropriate estimates. Assume that x f (x, u) is C 1+ and take ( x( ), ū( ) ) optimal. Use a priori estimates on the ODEs { ẋ ± (s) = f ( x ± (s), ū(s) ) a.e. s [t, T ] (ODE ± ) x ± (t) = x ± z. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

22 x S

23 S { x(t) = f ( x(t), ū(t) ) x(0) = x Optimal x

24 S x + z { x(t) = f ( x(t), ū(t) ) x(0) = x Optimal x x z

25 S { ẋ+(t) = f ( x +(t), ū(t) ) x +(0) = x + z Same control ū( ) x + z { x(t) = f ( x(t), ū(t) ) x(0) = x Optimal x { ẋ (t) = f ( x (t), ū(t) ) x (0) = x z Same control ū( ) x z Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

26 This proof depends on the parameterization A priori estimates rely on the specific parameterization (x, u) f (x, u) of the dynamics, but the value functions T ( ) and V (, ) do not. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

27 This proof depends on the parameterization A priori estimates rely on the specific parameterization (x, u) f (x, u) of the dynamics, but the value functions T ( ) and V (, ) do not. Important: To obtain second order (SCC) estimates, one definitely requires more than mere Lipschitz of the map x f (x, u). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

28 This proof depends on the parameterization A priori estimates rely on the specific parameterization (x, u) f (x, u) of the dynamics, but the value functions T ( ) and V (, ) do not. Important: To obtain second order (SCC) estimates, one definitely requires more than mere Lipschitz of the map x f (x, u). The question is What? Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

29 This proof depends on the parameterization A priori estimates rely on the specific parameterization (x, u) f (x, u) of the dynamics, but the value functions T ( ) and V (, ) do not. Important: To obtain second order (SCC) estimates, one definitely requires more than mere Lipschitz of the map x f (x, u). The question is What? Recall the previous work assumed x f (x, u) is C 1+. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

30 (Very) simple example Note that T ( ) and V (, ) depend only on the trajectories x( ) and NOT in the parameterization of the admissible velocity set: F (x) := { f (x, u) : u U }. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

31 (Very) simple example Note that T ( ) and V (, ) depend only on the trajectories x( ) and NOT in the parameterization of the admissible velocity set: F (x) := { f (x, u) : u U }. Note the[ admissible ] velocity multifunction F : R R given by F (x) = x, x can be parameterized two ways: F (x) = { {x u : u 1} { x u : u 1} The former is smoothly parameterized whereas the latter is not. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

32 (Very) simple example Note that T ( ) and V (, ) depend only on the trajectories x( ) and NOT in the parameterization of the admissible velocity set: F (x) := { f (x, u) : u U }. Note the[ admissible ] velocity multifunction F : R R given by F (x) = x, x can be parameterized two ways: F (x) = { {x u : u 1} { x u : u 1} The former is smoothly parameterized whereas the latter is not. Trajectories coincide, but theorems only apply to the former. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

33 Differential Inclusions and Filippov s Lemma The set of solutions to the Differential Inclusion { ẋ(s) F ( x(s) ) a.e. s [t, T ] (DI) x(t) = x does not depend on the particular parameterization. This is (essentially) the content of the well-known Filippov s Lemma. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

34 Differential Inclusions and Filippov s Lemma The set of solutions to the Differential Inclusion { ẋ(s) F ( x(s) ) a.e. s [t, T ] (DI) x(t) = x does not depend on the particular parameterization. This is (essentially) the content of the well-known Filippov s Lemma. Natural question: For which F is the value function semiconcave? Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

35 Differential Inclusions and Filippov s Lemma The set of solutions to the Differential Inclusion { ẋ(s) F ( x(s) ) a.e. s [t, T ] (DI) x(t) = x does not depend on the particular parameterization. This is (essentially) the content of the well-known Filippov s Lemma. Natural question: For which F is the value function semiconcave? A satisfactory answer should be given in terms of F, or equivalently, by the Hamiltonian H : R n R n R defined by: H(x, p) = v, p. sup v F (x) Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

36 Equivalence of F and H We assume throughout that F : R n R n satisfies the following Standard Hypotheses: 1) F (x) is nonempty, convex, and compact x, (SH) + 2) F is Lipschitz on bounded sets w.r.t. Hausdorff metric; 3) r > 0 so that max{ v : v F (x)} r(1 + x ). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

37 Equivalence of F and H We assume throughout that F : R n R n satisfies the following Standard Hypotheses: 1) F (x) is nonempty, convex, and compact x, (SH) + 2) F is Lipschitz on bounded sets w.r.t. Hausdorff metric; 3) r > 0 so that max{ v : v F (x)} r(1 + x ). Such assumptions on F give way to equivalent conditions on H because v F (x) v, p H(x, p) p R n Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

38 Equivalence of F and H We assume throughout that F : R n R n satisfies the following Standard Hypotheses: 1) F (x) is nonempty, convex, and compact x, (SH) + 2) F is Lipschitz on bounded sets w.r.t. Hausdorff metric; 3) r > 0 so that max{ v : v F (x)} r(1 + x ). Such assumptions on F give way to equivalent conditions on H because v F (x) v, p H(x, p) p R n 1) x R n, H(x, p) is finite and convex, positively homogeneous in p; (SH) + 2) M > 0, k > 0 so that x, y M, p R n, H(x, p) H(y, p) k p x y ; 3) r > 0 so that H(x, p) r p (1 + x ) x, p R n. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

39 Smooth parameterizations? Perhaps one can characterize those multifunctions that have a smooth parameterization: Question: Given F : R n R n, when does there exist f : R n U R n that is C 1 in the first coordinate and satisfies F (x) := { f (x, u) : u U }? Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

40 Smooth parameterizations? Perhaps one can characterize those multifunctions that have a smooth parameterization: Question: Given F : R n R n, when does there exist f : R n U R n that is C 1 in the first coordinate and satisfies F (x) := { f (x, u) : u U }? This seems virtually impossible to answer. Worse: Even sufficient conditions for smooth selections seems intractable: A simpler (?) question: Under what conditions on F does there exist a C 1 function f : R n R n so that f (x) F (x) x? Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

41 Necessary conditions for smooth parameterizations We can say when a smooth parameterization does NOT exist! Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

42 Necessary conditions for smooth parameterizations We can say when a smooth parameterization does NOT exist! Suppose F : R n R n is smoothly parameterized and H has the form { } f H(x, p) = sup (x, u), p : u U, where f : R n U R n has f and f x both continuous in (x, u). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

43 Necessary conditions for smooth parameterizations We can say when a smooth parameterization does NOT exist! Suppose F : R n R n is smoothly parameterized and H has the form { } f H(x, p) = sup (x, u), p : u U, where f : R n U R n has f and f x Then for 0 p R n, we have (H1) The map x H(x, p) is semiconvex; both continuous in (x, u). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

44 Necessary conditions for smooth parameterizations We can say when a smooth parameterization does NOT exist! Suppose F : R n R n is smoothly parameterized and H has the form { } f H(x, p) = sup (x, u), p : u U, where f : R n U R n has f and f x Then for 0 p R n, we have (H1) The map x H(x, p) is semiconvex; and (NC) If H(x, p) = H(x, p), then both continuous in (x, u). x H(x, p) = x H(x, p). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

45 Necessary conditions for smooth parameterizations We can say when a smooth parameterization does NOT exist! Suppose F : R n R n is smoothly parameterized and H has the form { } f H(x, p) = sup (x, u), p : u U, where f : R n U R n has f and f x Then for 0 p R n, we have (H1) The map x H(x, p) is semiconvex; and (NC) If H(x, p) = H(x, p), then both continuous in (x, u). x H(x, p) = x H(x, p). If the assumption of a smooth parameterization is replaced by the existence of a smooth selection, then the conclusion of (NC) is x H(x, p) x H(x, p). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

46 Illustration of (NC) with n = 1 F (x) H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

47 Illustration of (NC) with n = 1 F (x) F (x) = [ h(x), H(x) ] H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

48 Illustration of (NC) with n = 1 F (x) F (x) = [ h(x), H(x) ] H(x) F (x 3) F (x) x 1 x x 2 h(x) Graph of F (x) x x 3 x 4 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

49 Illustration of (NC) with n = 1 F (x) no smooth parameterization H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 x 1 : No smooth parameterization since x H(x) not semiconvex. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

50 Illustration of (NC) with n = 1 F (x) smooth parameterizations H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 x 2, x 3 : Smooth parameterizations are possible between x 1 and x 4. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

51 Illustration of (NC) with n = 1 F (x) smooth parameterizations H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

52 Illustration of (NC) with n = 1 F (x) smooth parameterizations H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

53 Illustration of (NC) with n = 1 F (x) smooth parameterizations H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

54 Illustration of (NC) with n = 1 F (x) smooth parameterizations H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 x 2, x 3 : Smooth parameterizations are possible between x 1 and x 4. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

55 Illustration of (NC) with n = 1 F (x) smooth parameterizations H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 x 2, x 3 : Smooth parameterizations are possible between x 1 and x 4. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

56 Illustration of (NC) with n = 1 F (x) H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

57 Illustration of (NC) with n = 1 F (x) no smooth selection H(x) x 1 x 2 h(x) Graph of F (x) x x 3 x 4 x 4 : No smooth selection. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

58 Proof of (NC) Proof. Note that the assumption means that H(x, p) = H(x, p) sup f (x, u), p = sup v, p = inf v, p = inf f (x, u), p u U v F (x) v F (x) u U That is, the assumption is that every u U both minimizes and maximizes the quantity f (x, u), p. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

59 Proof of (NC) Proof. Note that the assumption means that H(x, p) = H(x, p) sup f (x, u), p = sup v, p = inf v, p = inf f (x, u), p u U v F (x) v F (x) u U That is, the assumption is that every u U both minimizes and maximizes the quantity f (x, u), p. By a theorem on nonsmooth differentiation of max functions, one has { } x H(x, p) = co x f (x, u) p : u U and { } x H(x, p) = co x f (x, u) p : u U, from which (NC) follows: Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

60 Proof of (NC) Proof. Note that the assumption means that H(x, p) = H(x, p) sup f (x, u), p = sup v, p = inf v, p = inf f (x, u), p u U v F (x) v F (x) u U That is, the assumption is that every u U both minimizes and maximizes the quantity f (x, u), p. By a theorem on nonsmooth differentiation of max functions, one has { } x H(x, p) = co x f (x, u) p : u U and { } x H(x, p) = co x f (x, u) p : u U, from which (NC) follows: x H(x, p) = x H(x, p). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

61 New DI assumptions We abandon looking for smooth parameterizations, and introduce: { 1) x H(x, p) is semiconvex, and (H) 2) The gradient p H(x, p) exists and is locally Lipschitz in x. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

62 New DI assumptions We abandon looking for smooth parameterizations, and introduce: { 1) x H(x, p) is semiconvex, and (H) 2) The gradient p H(x, p) exists and is locally Lipschitz in x. Class of examples: One can generate a class of examples that satisfy (H) but do not satisfy (NC), and therefore could not have a C 1 parameterization: Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

63 New DI assumptions We abandon looking for smooth parameterizations, and introduce: { 1) x H(x, p) is semiconvex, and (H) 2) The gradient p H(x, p) exists and is locally Lipschitz in x. Class of examples: One can generate a class of examples that satisfy (H) but do not satisfy (NC), and therefore could not have a C 1 parameterization: n = 1: Let F (x) := [ h(x), H(x) ] where h( ) and H( ) are semiconvex. These always satisfy (H), and could satisfy (NC) only if h(x) = H(x) h(x) = H(x). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

64 New DI assumptions We abandon looking for smooth parameterizations, and introduce: { 1) x H(x, p) is semiconvex, and (H) 2) The gradient p H(x, p) exists and is locally Lipschitz in x. Class of examples: One can generate a class of examples that satisfy (H) but do not satisfy (NC), and therefore could not have a C 1 parameterization: n = 1: Let F (x) := [ h(x), H(x) ] where h( ) and H( ) are semiconvex. These always satisfy (H), and could satisfy (NC) only if h(x) = H(x) h(x) = H(x). n > 1: Let F (x) := f (x) + r(x)b where f ( ) is C 2 and r : R n [0, ) is semiconvex. Then H(x, p) = f (x), p + r(x) p, and so (H) is satisfied. Then (NC) is satisfied only if r(x) = 0 implies r(x) = r(x). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

65 Consequences, part I Consequences of (H1): The semiconvexity of x H(x, p) implies x,p H(x, p) x H ( x, p ) p H ( x, p ). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

66 Consequences, part I Consequences of (H1): The semiconvexity of x H(x, p) implies x,p H(x, p) x H ( x, p ) p H ( x, p ). The significance of this result is in utilizing a nonsmooth maximum principle (Clarke 1975): Suppose x( ) is optimal in one of the classical problems with (DI) dynamics. Then there exists an adjoint arc p( ) for which ( p(s), x(s) ) x,p H ( x(s), p(s) ) (plus transversality conditions). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

67 Consequences, part I Consequences of (H1): The semiconvexity of x H(x, p) implies x,p H(x, p) x H ( x, p ) p H ( x, p ). The significance of this result is in utilizing a nonsmooth maximum principle (Clarke 1975): Suppose x( ) is optimal in one of the classical problems with (DI) dynamics. Then there exists an adjoint arc p( ) for which ( p(s), x(s) ) x,p H ( x(s), p(s) ) (plus transversality conditions). Thus the dynamics of a Hamiltonian arc ( x( ), p( ) ) splits into a much more usable form: p(s) x H ( x(s), p(s) ) and x(s) p H ( x(s), p(s) ) Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

68 Consequences, part II Consequences of (H2): That the gradient p H(x, p) exists means that the argmax of v, p is unique - we denote it by f p (x) F (x). sup v F (x) Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

69 Consequences, part II Consequences of (H2): That the gradient p H(x, p) exists means that the argmax of v, p is unique - we denote it by f p (x) F (x). sup v F (x) We have also assumed x f p (x) is Lipschitz. If p( ) is any continuous function, then the ODE (ODE) x { ẋ(t) = f p(t) ( x(t) ) x(0) = x, satisfies standard Carathéodory-type assumptions a.e. t [0, T ] Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

70 Consequences, part II Consequences of (H2): That the gradient p H(x, p) exists means that the argmax of v, p is unique - we denote it by f p (x) F (x). sup v F (x) We have also assumed x f p (x) is Lipschitz. If p( ) is any continuous function, then the ODE (ODE) x { ẋ(t) = f p(t) ( x(t) ) x(0) = x, a.e. t [0, T ] satisfies standard Carathéodory-type assumptions, and has the properties: A unique solution x( ; x) of (ODE) x exists on [0, ); Each solution x( ; x) is a solution of (DI); The function x x(t; x) is locally Lipschitz; If ( x( ), p( ) ) is a Hamiltonian arc, then x( ) satisfies (ODE) with p( ) = p( ). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

71 A replacement for a priori estimates S x

72 A replacement for a priori estimates S { x(t) = f p(t) ( x(t) ) x(0) = x Optimal with adjoint p( ) x

73 A replacement for a priori estimates S x + z { x(t) = f p(t) ( x(t) ) x(0) = x Optimal with adjoint p( ) x x z

74 A replacement for a priori estimates S { ẋ+(t) = f p(t) ( x+(t) ) x +(0) = x + z Same adjoint arc p( ) x + z { x(t) = f p(t) ( x(t) ) x(0) = x Optimal with adjoint p( ) x { ẋ (t) = f p(t) ( x (t) ) x (0) = x z Same adjoint arc p( ) x z Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

75 Summary of new results We assume F satisfies (SH) + and (H). Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

76 Summary of new results We assume F satisfies (SH) + and (H). Theorem Consider the minimum time problem. Suppose the target S is compact and satisfies the Petrov condition and the Interior Sphere Property. Then there exists ρ > 0 so that T ( ) is semiconcave on each convex subset of S + ρb \ S. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

77 Summary of new results We assume F satisfies (SH) + and (H). Theorem Consider the minimum time problem. Suppose the target S is compact and satisfies the Petrov condition and the Interior Sphere Property. Then there exists ρ > 0 so that T ( ) is semiconcave on each convex subset of S + ρb \ S. Theorem Consider the Mayer problem. Assume the endpoint cost function l( ) is semiconcave. Then the value function V (, ) is locally semiconcave on (, T ] R n. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

78 Conclusions and future work General Philosophy: Assumptions should not be required only in the proof. Here, smooth parameterizations of control systems should not be assumed if the conclusion does not reflect them. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

79 Conclusions and future work General Philosophy: Assumptions should not be required only in the proof. Here, smooth parameterizations of control systems should not be assumed if the conclusion does not reflect them. We offered an alternative; It is not yet clear how generic these assumptions are. For example, could they be invoked in an auxiliary problem that approximates an original one? Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

80 Conclusions and future work General Philosophy: Assumptions should not be required only in the proof. Here, smooth parameterizations of control systems should not be assumed if the conclusion does not reflect them. We offered an alternative; It is not yet clear how generic these assumptions are. For example, could they be invoked in an auxiliary problem that approximates an original one? Local versions? Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

81 Conclusions and future work General Philosophy: Assumptions should not be required only in the proof. Here, smooth parameterizations of control systems should not be assumed if the conclusion does not reflect them. We offered an alternative; It is not yet clear how generic these assumptions are. For example, could they be invoked in an auxiliary problem that approximates an original one? Local versions? Perhaps a large segment of optimal control can be cast in this way.(?) Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

82 Conclusions and future work General Philosophy: Assumptions should not be required only in the proof. Here, smooth parameterizations of control systems should not be assumed if the conclusion does not reflect them. We offered an alternative; It is not yet clear how generic these assumptions are. For example, could they be invoked in an auxiliary problem that approximates an original one? Local versions? Perhaps a large segment of optimal control can be cast in this way.(?) Students: you re urged and invited to join the fun. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

83 Conclusions and future work General Philosophy: Assumptions should not be required only in the proof. Here, smooth parameterizations of control systems should not be assumed if the conclusion does not reflect them. We offered an alternative; It is not yet clear how generic these assumptions are. For example, could they be invoked in an auxiliary problem that approximates an original one? Local versions? Perhaps a large segment of optimal control can be cast in this way.(?) Students: you re urged and invited to join the fun. Thank you for your attention and for sticking around. Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

84 Thank you for your attention! Grazie per l attenzione! Dziękuję za uwagę! Merci de l attention Obrigado pela atençāo Vielen Dank für Ihre Aufmerksamkeit Gracias por su atención Gràcies per la vostra atenció Cám on Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

85 And finally... Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

86 And finally... Many thanks to Richard, Rosario, Hasnaa, Estelle, and all the SADCO people! Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22

87 And finally... Many thanks to Richard, Rosario, Hasnaa, Estelle, and all the SADCO people! It s been a great week! Peter R. Wolenski (LSU) Semiconcavity and control SADCO summer / 22