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Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Introduction to Differential Inclusions September 26, 2007 Rick Barnard Student Seminar on Control Theory and Optimization Fall 2007 Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions 1 Definitions 2 Selections 3 Differential Inclusions Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions References Aubin, J.P. and Cellina, A. Differential Inclusions. Springer-Verlag, Berlin, 1984. Clarke, F.H. Optimization and Nonsmoorth Analysis. Centre de Recherches Mathe´matiques, Montre´al, 1989. Smirnov, G.S. Introduction to the Theory of Differential Inclusions. AMS, Providence, 2002. C. Cai, R. Goebel, R. Sanfelice, A. Teel. Lecture Notes to Workshop on Robust Hybrid Systems: Theory and Applications. CDc ’06. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Multifunctions A multifunction F : Rm → Rn is a map from Rm to the subsets of Rn, that is for every x ∈ Rm, we associate a (potentially empty) set F (x). Its graph, denoted Gr(F ) is defined by Gr(F ) = {(x , y)|y ∈ F (x)}. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Measurability A multifunction F : S → Rn is measurable if for every open (closed) C ⊆ Rn, {x ∈ S : F (x) ∩ C 6= ∅} is Lebesgue measurable. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Continuity A multifunction F is called upper semi-continuous at x0 if for any open M containing F (x0) there is a neighborhood Ω of x0 so that F (Ω) ⊂ M. A multifunction F is called lower semi-continuous at x0 if for any y0 ∈ F (x0) and any neighborhood M of y0 there is a negihborhood Ω of x0 so that F (x) ∩M 6= ∅, ∀x ∈ Ω. A multifunction is continuous at x0 if it is both upper and lower semi-continuous at x0. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Continuity A multifunction F is called upper semi-continuous at x0 if for any open M containing F (x0) there is a neighborhood Ω of x0 so that F (Ω) ⊂ M. A multifunction F is called lower semi-continuous at x0 if for any y0 ∈ F (x0) and any neighborhood M of y0 there is a negihborhood Ω of x0 so that F (x) ∩M 6= ∅, ∀x ∈ Ω. A multifunction is continuous at x0 if it is both upper and lower semi-continuous at x0. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Lipschitz Continuity A multifunction F is said to be Lipschitz continuous if there is a k ≥ 0 so that for any x1, x2 ∈ Rm we have F (x1) ⊂ F (x2) + k|x1 − x2|B. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions The selection problem Given a multifunction F : Rm → Rn, a single-valued map f : Rm → Rn is a selection if f (x) ∈ F (x), ∀x ∈ Rm. For what multifunctions are we assured of the existence of a selection? Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions The selection problem Given a multifunction F : Rm → Rn, a single-valued map f : Rm → Rn is a selection if f (x) ∈ F (x), ∀x ∈ Rm. For what multifunctions are we assured of the existence of a selection? Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Michael’s Selection Theorem and a Measurable Selection Theorem Theorem Let F be a closed, convex, and lower semi-continuous multifunction. Then there is a continuous selection from F . Theorem Let F be measurable, closed, and nonempty on S . Then there is a measurable selection from F . Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Approximate Selections Theorem Let F be a convex, upper semi-continuous multifunction. Then for ² > 0 there is a locally Lipschitz continuous function f² whose range is in the convex hull of the range of F and Gr(f²) ⊂ Gr(F ) + ²B. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Differential Inclusions We now turn our attention to the problem of solving Differential Inclusions: x˙(t) ∈ F (x(t)), t ∈ [0,T ], (1) with x(0) = x0. We will assume that F is closed, convex, and Lipschitz continuous with constant k > 0. We shall see presently that these assumptions are not too restrictive, especially when we concern ourselves with problems from control theory. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions An Example from Control Theory Consider the control system x˙ = f (x , u) where u ∈ U ⊂ Rk . We assume that f is Lipschitz in x , and we allow any measurable functions u : [0, t]→ Rk so that u(t) ∈ U a.e. and that for all x , f (x , u(t)) ∈ L1([0,T ]). Consider the multifunction F defined by F (x) = f (x ,U) = ⋃ u∈U f (x , u). We do the same for nonautonomous systems using the right-hand side F (t, x(t)). Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions An Example from Control Theory Consider the control system x˙ = f (x , u) where u ∈ U ⊂ Rk . We assume that f is Lipschitz in x , and we allow any measurable functions u : [0, t]→ Rk so that u(t) ∈ U a.e. and that for all x , f (x , u(t)) ∈ L1([0,T ]). Consider the multifunction F defined by F (x) = f (x ,U) = ⋃ u∈U f (x , u). We do the same for nonautonomous systems using the right-hand side F (t, x(t)). Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Filippov’s Theorem Theorem Let f : Rm × Rk → Rn be continuous, and let v : Rm → Rn be measurable. Assume U is compact so that v(x) ∈ f (x ,U) a.e. Then there is a measurable u : Rm → U satisfying v(x) = f (x , u(x)). Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Filippov’s Theorem Sketch of Proof: We know that U(x) = {u ∈ U| v(x) = f (x , u)} has compact values. Let u(x) = (u1(x), . . . , uk(x)) ∈ U(x) with u1(x) the smallest possible. We show that if ui (x) is measurable on a compact set A for i < p then up(x) is as well. Then we use the Lusin theorem to find a set A² where the ui (x)’s and v(x) are continuous and m(A \ A²) ≤ ². We then show that the sublevel sets of up(x) restricted to A² are closed. So up(x) is measurable on A² and we then repeat Lusin on this set to get an A2². Since ² is arbitrary we use Lusin again and see it is measurable on A. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Filippov’s Theorem Theorem Let f : Rm × Rk → Rn be continuous, and let v : Rm → Rn be measurable. Assume U is compact so that v(x) inf(x ,U) a.e. Then there is a measurable u : Rm → U satisfying v(x) = f (x , u(x)). Obviously, trajectories of the control system are solutions to the differential inclusion. This result means the converse holds and so the system and the inclusion are equivalent. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Continuity and Relaxation of Differential Inclusions Proposition Let U be compact and f : Rm × U → Rn be continuous (Lipschitz) in x . Then F : x → f (x ,U) is continuous (Lipschitz). Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Continuity and Relaxation of Differential Inclusions Proof. We know that F has closed graph and is locally bounded. Let N be a neighborhood of F (x) with sequences {xi}, {yi} so that xi → x0, yi ∈ F (xi ), and yi /∈ N. Since cl{F (x)| |x − x0| < δ} is compact, WLOG assume yi → y0. But the graph is closed, so y0 ∈ F (x0). So F is upper semi-continuous. The lower semi-continuity follows from the continuity of x → f (x , u), ∀u ∈ U. The Lipschitz statements are similarly straightforward. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Continuity and Relaxation of Differential Inclusions A trajectory is an absolutely continuous function x : [a, b]→ Rn such that x˙(t) ∈ F (t, x(t)) a.e. We say F is integrably bounded if there is an integrable function φ(·) such that |v | ≤ φ(t) for all v in F (t, x). Proposition A relaxed trajectory y is a trajectory for coF . That is, y˙(t) ∈ coF (t, y(t)). If F is integrably bounded then any relaxed trajectory y is within δ of a trajectory for F in the sup norm. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Continuity and Relaxation of Differential Inclusions Proposition Let U be compact and f : Rm × U → Rn be continuous (Lipschitz) in x . Then F : x → f (x ,U) is continuous (Lipschitz). Proposition A relaxed trajectory is a trajectory for coF . If F is integrably bounded then any relaxed trajectory y is within δ of a trajectory for F in the sup norm. Introduction to Differential Inclusions Introduction to Differential Inclusions Definitions Selections Differential Inclusions Definitions Selections Differential Inclusions Existence Theorem Theorem Assume that F (x) is closed, convex, and Lipschitzian. For any x0 ∈ Rn there exist solutions to (1) with x(0) = x0. Introduction to Differential Inclusions Definitions Selections Differential Inclusions