Moreau’s Sweeping Process and its control Giovanni Colombo Universit` a di Padova June 15, 2016 Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 1 / 35
Joint works with – B. Mordukhovich (Wayne State), Nguyen D. Hoang (Valparaiso, Chile), R. Henrion (WIAS) – Michele Palladino (Penn State) – Chems Eddine Arroud (Jijel, Algeria) Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 2 / 35
Outline of the Talk The dynamics of the sweeping process Control problems for the sweeping process A Hamilton-Jacobi characterization of the minimum time function Necessary optimality conditions Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 3 / 35
The dynamics of the sweeping process Consider a moving set C ( t ), depending on the time t ∈ [0 , T ], and an initial condition x 0 ∈ C (0). In several contexts, the modelization of the displacement x ( t ) of the initial condition x 0 subject to the dragging, or sweeping due to the displacement of C ( t ) pops up. It is natural to think that the point x ( t ) remains at rest until it is caught by the boundary of C ( t ) and then its velocity is normal to ∂ C ( t ). It is a kind of one sided movement. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 4 / 35
Formally, the sweeping process ( processus de rafle ) is the differential inclusion with initial condition x ( t ) ∈ − N C ( t ) ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) . Here N C ( x ) denotes the normal cone to C at x ∈ C . In particular, = { 0 } if x ∈ intC N C ( x ) N C ( x ) = ∅ if x / ∈ C . Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 5 / 35
Formally, the sweeping process ( processus de rafle ) is the differential inclusion with initial condition x ( t ) ∈ − N C ( t ) ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) . Here N C ( x ) denotes the normal cone to C at x ∈ C . In particular, = { 0 } if x ∈ intC N C ( x ) N C ( x ) = ∅ if x / ∈ C . The simplest example is the play operator: x ( t ) ∈ − N C + u ( t ) ( x ( t )) , ˙ namely C ( t ) is a translation. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 5 / 35
Classical assumptions are: x ∈ H , a Hilbert space t �→ C ( t ) is Lipschitz continuous C ( t ) is closed and convex Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35
Classical assumptions are: x ∈ H , a Hilbert space t �→ C ( t ) is Lipschitz continuous C ( t ) is closed and convex Classical results are (the inclusion is forward in time): Existence of a Lipschitz solution for the Cauchy problem Uniqueness and continuous dependence from the initial condition Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35
Classical assumptions are: x ∈ H , a Hilbert space t �→ C ( t ) is Lipschitz continuous C ( t ) is closed and convex Classical results are (the inclusion is forward in time): Existence of a Lipschitz solution for the Cauchy problem Uniqueness and continuous dependence from the initial condition Classical methods are: Moreau-Yosida approximation The catching up algorithm (both constructing a Cauchy sequence of approximate solutions). This goes back to J.-J. Moreau (early ’70s). Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35
Classical assumptions are: x ∈ H , a Hilbert space t �→ C ( t ) is Lipschitz continuous C ( t ) is closed and convex Classical results are (the inclusion is forward in time): Existence of a Lipschitz solution for the Cauchy problem Uniqueness and continuous dependence from the initial condition Classical methods are: Moreau-Yosida approximation The catching up algorithm (both constructing a Cauchy sequence of approximate solutions). This goes back to J.-J. Moreau (early ’70s). There are also (too) many generalizations. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35
The perturbed sweeping process: x ( t ) ∈ − N C ( t ) ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) . (1) Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35
The perturbed sweeping process: x ( t ) ∈ − N C ( t ) ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) . (1) If f is Lipschitz, essentially the above results and methods are valid. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35
The perturbed sweeping process: x ( t ) ∈ − N C ( t ) ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) . (1) If f is Lipschitz, essentially the above results and methods are valid. The important case where C is constant, namely x ( t ) ∈ − N C ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) , (2) models a state constrained evolution where the constraint appears actively in the dynamics. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35
The perturbed sweeping process: x ( t ) ∈ − N C ( t ) ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) . (1) If f is Lipschitz, essentially the above results and methods are valid. The important case where C is constant, namely x ( t ) ∈ − N C ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) , (2) models a state constrained evolution where the constraint appears actively in the dynamics. It is different from weak flow invariance, a.k.a. viability, because the state constraint is built in the dynamics. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35
The perturbed sweeping process: x ( t ) ∈ − N C ( t ) ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) . (1) If f is Lipschitz, essentially the above results and methods are valid. The important case where C is constant, namely x ( t ) ∈ − N C ( x ( t )) + f ( x ( t )) , ˙ x (0) = x 0 ∈ C (0) , (2) models a state constrained evolution where the constraint appears actively in the dynamics. It is different from weak flow invariance, a.k.a. viability, because the state constraint is built in the dynamics. Note that the right hand side of (1), (2) is discontinuous with respect to the state, for two different reasons. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35
A proof of continuous dependence from the initial condition. Two solutions: x 1 ( t ) ∈ − N C ( t ) ( x 1 ( t )) + f ( x 1 ( t )) , ˙ x 1 (0) = ξ 1 x 2 ( t ) ∈ − N C ( t ) ( x 2 ( t )) + f ( x 2 ( t )) , ˙ x 2 (0) = ξ 2 Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35
A proof of continuous dependence from the initial condition. Two solutions: x 1 ( t ) ∈ − N C ( t ) ( x 1 ( t )) + f ( x 1 ( t )) , ˙ x 1 (0) = ξ 1 x 2 ( t ) ∈ − N C ( t ) ( x 2 ( t )) + f ( x 2 ( t )) , ˙ x 2 (0) = ξ 2 By convexity: � � � � − ˙ x 1 ( t ) + f ( x 1 ( t )) · x 2 ( t ) − x 1 ( t ) ≤ 0 � � � � − ˙ · x 1 ( t ) − x 2 ( t ) ≤ 0 x 2 ( t ) + f ( x 2 ( t )) Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35
A proof of continuous dependence from the initial condition. Two solutions: x 1 ( t ) ∈ − N C ( t ) ( x 1 ( t )) + f ( x 1 ( t )) , ˙ x 1 (0) = ξ 1 x 2 ( t ) ∈ − N C ( t ) ( x 2 ( t )) + f ( x 2 ( t )) , ˙ x 2 (0) = ξ 2 By convexity: � � � � − ˙ x 1 ( t ) + f ( x 1 ( t )) · x 2 ( t ) − x 1 ( t ) ≤ 0 � � � � − ˙ · x 1 ( t ) − x 2 ( t ) ≤ 0 x 2 ( t ) + f ( x 2 ( t )) Summing and rearranging: � x 2 ( t ) − ˙ � · � x 2 ( t ) − x 1 ( t ) � ≤ � f ( x 2 ( t )) − f ( x 1 ( t )) � · � x 2 ( t ) − x 1 ( t ) � ˙ x 1 ( t ) � 2 � ≤ L � x 2 ( t ) − x 1 ( t ) Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35
A proof of continuous dependence from the initial condition. Two solutions: x 1 ( t ) ∈ − N C ( t ) ( x 1 ( t )) + f ( x 1 ( t )) , ˙ x 1 (0) = ξ 1 x 2 ( t ) ∈ − N C ( t ) ( x 2 ( t )) + f ( x 2 ( t )) , ˙ x 2 (0) = ξ 2 By convexity: � � � � − ˙ x 1 ( t ) + f ( x 1 ( t )) · x 2 ( t ) − x 1 ( t ) ≤ 0 � � � � − ˙ · x 1 ( t ) − x 2 ( t ) ≤ 0 x 2 ( t ) + f ( x 2 ( t )) Summing and rearranging: � x 2 ( t ) − ˙ � · � x 2 ( t ) − x 1 ( t ) � ≤ � f ( x 2 ( t )) − f ( x 1 ( t )) � · � x 2 ( t ) − x 1 ( t ) � ˙ x 1 ( t ) � 2 � ≤ L � x 2 ( t ) − x 1 ( t ) Conclude by Gronwall’s lemma. Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35
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