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Sparse Jurdjevic-Quinn stabilization Francesco Rossi - Universit - PowerPoint PPT Presentation

Oct 30, 2023 •279 likes •1.4k views

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Sparse Jurdjevic-Quinn stabilization Francesco Rossi - Universit dAix-Marseille, France http://www.LSIS.org/frossi/

  1. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations The simplest example on R 2 � ˙ � 1 � 0 � � � x = u 1 g 1 + u 2 g 2 = u 1 + u 2 y ˙ 0 1 Exercise One can stabilize the system to (0 , 0) by choosing the functional V = x 2 + y 2 . f = 0 , hence L f V = 0 ; L g 1 V = L g 2 V = 0 in (0 , 0) only; Jurdjevic-Quinn gives u ( x, y ) = ( − 2 x, − 2 y ) , What happens when adding the component-wise sparsity constraint? 9 / 40

  2. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Hybrid systems to prevent chattering The component-wise sparsity constraint itself produces switching rules, that naturally introduce chattering phenomena. We present three possible solutions: Strategies 1 and 2 are based on time sampling; Strategy 3 is based on hysteresis. 10 / 40

  3. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Hybrid systems to prevent chattering The component-wise sparsity constraint itself produces switching rules, that naturally introduce chattering phenomena. We present three possible solutions: Strategies 1 and 2 are based on time sampling; Strategy 3 is based on hysteresis. 10 / 40

  4. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 11 / 40

  5. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 11 / 40

  6. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback Fix the sampling time τ > 0 . For the initial state x 0 ∈ R n , consider the trajectory x ( t ) of (S) with the time-varying feedback control u ( t, x ) defined as follows: on each time interval [( km + i ) τ, ( km + i + 1) τ ) , apply the feedback control u ( t, x ) = − L g i V ( x ( t )) e i . Remark This is a kind of averaged control. 12 / 40

  7. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback Fix the sampling time τ > 0 . For the initial state x 0 ∈ R n , consider the trajectory x ( t ) of (S) with the time-varying feedback control u ( t, x ) defined as follows: on each time interval [( km + i ) τ, ( km + i + 1) τ ) , apply the feedback control u ( t, x ) = − L g i V ( x ( t )) e i . Remark This is a kind of averaged control. 12 / 40

  8. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback Theorem (Caponigro-Piccoli-Rossi-Trélat, submitted) Let Z = { x ∈ R n | L f V ( x ) = L g i V ( x ) = 0 , for every i = 1 , . . . , m } , and Ω be the largest subset of Z invariant for ˙ x = f ( x ) . If Ω is locally attractive , then, for each initial state x 0 ∈ R n , there exist τ 1 > 0 such that, for every τ ∈ (0 , τ 1 ) , Strategy 1 with sampling time τ stabilizes the control system (S) to Ω . Remark Local attractiveness is necessary, see counterexamples. 13 / 40

  9. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback Theorem (Caponigro-Piccoli-Rossi-Trélat, submitted) Let Z = { x ∈ R n | L f V ( x ) = L g i V ( x ) = 0 , for every i = 1 , . . . , m } , and Ω be the largest subset of Z invariant for ˙ x = f ( x ) . If Ω is locally attractive , then, for each initial state x 0 ∈ R n , there exist τ 1 > 0 such that, for every τ ∈ (0 , τ 1 ) , Strategy 1 with sampling time τ stabilizes the control system (S) to Ω . Remark Local attractiveness is necessary, see counterexamples. 13 / 40

  10. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback τ = 1 14 / 40

  11. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback τ = 0 . 1 14 / 40

  12. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback τ = 0 . 01 14 / 40

  13. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback Strategy 1 is far from being “optimal” in some sense; 14 / 40

  14. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1: Sparse time-varying feedback Strategy 1 is far from being “optimal” in some sense; It is useful on long-range, not locally. 14 / 40

  15. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 15 / 40

  16. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 15 / 40

  17. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sampled sparse feedback Fix a sampling time τ > 0 . In x 0 , choose the smallest integer i ∈ { 1 , . . . , m } such that | L g i V ( x 0 ) | ≥ | L g j V ( x 0 ) | , ∀ j � = i. Then consider the sampling solution associated with u and the sampling time τ until x 1 = x ( τ ) . Compute the new control in x 1 , and interatively for each x n . Remark The control is chosen as a “steepest descent” for V , then it is kept for the whole time interval [0 , τ ] . 16 / 40

  18. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sampled sparse feedback Fix a sampling time τ > 0 . In x 0 , choose the smallest integer i ∈ { 1 , . . . , m } such that | L g i V ( x 0 ) | ≥ | L g j V ( x 0 ) | , ∀ j � = i. Then consider the sampling solution associated with u and the sampling time τ until x 1 = x ( τ ) . Compute the new control in x 1 , and interatively for each x n . Remark The control is chosen as a “steepest descent” for V , then it is kept for the whole time interval [0 , τ ] . 16 / 40

  19. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sampled sparse feedback Theorem (CPRT) If { x ∈ R n | L f V ( x ) = L g i V ( x ) = 0 , for every i = 1 , . . . , m } = { ¯ x } x ∈ R n , then, for every initial state x 0 ∈ R n , there exists for some ¯ τ 2 > 0 such that, for every τ ∈ (0 , τ 2 ) , Strategy 2 with sampling time τ practically asymptotically stabilizes (S) to ¯ x . Remark The functional V ( t ) is not strictly decreasing, one can have problems for Z not reduced to a point. 17 / 40

  20. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sampled sparse feedback Theorem (CPRT) If { x ∈ R n | L f V ( x ) = L g i V ( x ) = 0 , for every i = 1 , . . . , m } = { ¯ x } x ∈ R n , then, for every initial state x 0 ∈ R n , there exists for some ¯ τ 2 > 0 such that, for every τ ∈ (0 , τ 2 ) , Strategy 2 with sampling time τ practically asymptotically stabilizes (S) to ¯ x . Remark The functional V ( t ) is not strictly decreasing, one can have problems for Z not reduced to a point. 17 / 40

  21. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sparse time-varying feedback τ = 1 18 / 40

  22. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sparse time-varying feedback τ = 0 . 5 18 / 40

  23. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sparse time-varying feedback τ = 0 . 01 18 / 40

  24. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2: Sparse time-varying feedback Strategy 2 is efficient along axes. It is not efficient around the switching set. 18 / 40

  25. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Fix ε ∈ (0 , 1) and apply the following algorithm: Initialize: n = 0 and t 0 = 0 . While t n < + ∞ apply Step n : At time t n choose i = 1 , . . . , m the smallest integer such that | L g i V ( x ( t n )) | ≥ | L g j V ( x ( t n )) | , for every j � = i. If | L g i V ( x ( t n )) | ≥ 2 t − 1 n , keep the control u = − L g i V ( x ) until | L g i V ( x ( t )) | ≤ t − 1 or | L g i V x ( t )) | ≤ (1 − ε ) | L g j V ( x ( t )) | for some j = 1 , . . . , m. If | L g i V ( x ( t n )) | < 2 t − 1 n , keep the control u = 0 until | L g j V ( x ( t )) | ≥ 4 t − 1 , for some j = 1 , . . . , m. Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time. 19 / 40

  26. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Fix ε ∈ (0 , 1) and apply the following algorithm: Initialize: n = 0 and t 0 = 0 . While t n < + ∞ apply Step n : At time t n choose i = 1 , . . . , m the smallest integer such that | L g i V ( x ( t n )) | ≥ | L g j V ( x ( t n )) | , for every j � = i. If | L g i V ( x ( t n )) | ≥ 2 t − 1 n , keep the control u = − L g i V ( x ) until | L g i V ( x ( t )) | ≤ t − 1 or | L g i V x ( t )) | ≤ (1 − ε ) | L g j V ( x ( t )) | for some j = 1 , . . . , m. If | L g i V ( x ( t n )) | < 2 t − 1 n , keep the control u = 0 until | L g j V ( x ( t )) | ≥ 4 t − 1 , for some j = 1 , . . . , m. Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time. 19 / 40

  27. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Fix ε ∈ (0 , 1) and apply the following algorithm: Initialize: n = 0 and t 0 = 0 . While t n < + ∞ apply Step n : At time t n choose i = 1 , . . . , m the smallest integer such that | L g i V ( x ( t n )) | ≥ | L g j V ( x ( t n )) | , for every j � = i. If | L g i V ( x ( t n )) | ≥ 2 t − 1 n , keep the control u = − L g i V ( x ) until | L g i V ( x ( t )) | ≤ t − 1 or | L g i V x ( t )) | ≤ (1 − ε ) | L g j V ( x ( t )) | for some j = 1 , . . . , m. If | L g i V ( x ( t n )) | < 2 t − 1 n , keep the control u = 0 until | L g j V ( x ( t )) | ≥ 4 t − 1 , for some j = 1 , . . . , m. Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time. 19 / 40

  28. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Fix ε ∈ (0 , 1) and apply the following algorithm: Initialize: n = 0 and t 0 = 0 . While t n < + ∞ apply Step n : At time t n choose i = 1 , . . . , m the smallest integer such that | L g i V ( x ( t n )) | ≥ | L g j V ( x ( t n )) | , for every j � = i. If | L g i V ( x ( t n )) | ≥ 2 t − 1 n , keep the control u = − L g i V ( x ) until | L g i V ( x ( t )) | ≤ t − 1 or | L g i V x ( t )) | ≤ (1 − ε ) | L g j V ( x ( t )) | for some j = 1 , . . . , m. If | L g i V ( x ( t n )) | < 2 t − 1 n , keep the control u = 0 until | L g j V ( x ( t )) | ≥ 4 t − 1 , for some j = 1 , . . . , m. Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time. 19 / 40

  29. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Theorem (CPRT) For any ε ∈ (0 , 1) , Strategy 3 asymptotically stabilizes (S) to Ω , the largest invariant subset of { x ∈ R n | L f V ( x ) = L g i V ( x ) = 0 , for every i = 1 , . . . , m } . Remark The strategy is more robust, since it works for any ε . One needs a continuous evaluation of the control. 20 / 40

  30. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Theorem (CPRT) For any ε ∈ (0 , 1) , Strategy 3 asymptotically stabilizes (S) to Ω , the largest invariant subset of { x ∈ R n | L f V ( x ) = L g i V ( x ) = 0 , for every i = 1 , . . . , m } . Remark The strategy is more robust, since it works for any ε . One needs a continuous evaluation of the control. 20 / 40

  31. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Theorem (CPRT) For any ε ∈ (0 , 1) , Strategy 3 asymptotically stabilizes (S) to Ω , the largest invariant subset of { x ∈ R n | L f V ( x ) = L g i V ( x ) = 0 , for every i = 1 , . . . , m } . Remark The strategy is more robust, since it works for any ε . One needs a continuous evaluation of the control. 20 / 40

  32. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis ε = 0 . 5 21 / 40

  33. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis ε = 0 . 1 21 / 40

  34. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis ε = 0 . 01 21 / 40

  35. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3: Sparse feedback with hysteresis Similarly to Strategy 2, Strategy 3 is efficient along axes. It is not efficient around the switching set. 21 / 40

  36. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Comparison between strategies 8 6 4 2 0 0 2 4 6 8 Orange: “Time to target=1” for Strategy 1 for τ → 0 . Red: Strategy 2 for τ → 0 and Strategy 3 for ε → 0 . Green: Non-sparse Jurdjevic-Quinn. 22 / 40

  37. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Comparison between strategies 8 6 4 2 0 0 2 4 6 8 Orange: “Time to target=1” for Strategy 1 for τ → 0 . Red: Strategy 2 for τ → 0 and Strategy 3 for ε → 0 . Green: Non-sparse Jurdjevic-Quinn. 22 / 40

  38. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Sparse control of the Hegselmann-Krause model Each agent has an opinion x i ( t ) , that evolves by interaction with his neighbours only: � x i = ˙ ( x j − x i ) | x j − x i | < 1 23 / 40

  39. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Sparse control of the Hegselmann-Krause model Each agent has an opinion x i ( t ) , that evolves by interaction with his neighbours only: � x i = ˙ ( x j − x i ) + u i . | x j − x i | < 1 23 / 40

  40. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Sparse control of the Hegselmann-Krause model Each agent has an opinion x i ( t ) , that evolves by interaction with his neighbours only: � x i = ˙ ( x j − x i ) + u i . | x j − x i | < 1 We aim to enforce consensus by minimizing the variance � ( x i − x j ) 2 . V = i,j 23 / 40

  41. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Sparse control of the Hegselmann-Krause model Each agent has an opinion x i ( t ) , that evolves by interaction with his neighbours only: � x i = ˙ ( x j − x i ) + u i . | x j − x i | < 1 We aim to enforce consensus by minimizing the variance � ( x i − x j ) 2 . V = i,j L f V ≤ 0 , thus we can apply Jurdjevic-Quinn. 23 / 40

  42. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Simulation with 50 randomized agents in [0 , 10] 24 / 40

  43. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1 for Hegselmann-Krause model τ = 1 25 / 40

  44. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1 for Hegselmann-Krause model τ = 0 . 5 25 / 40

  45. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 1 for Hegselmann-Krause model τ = 0 . 1 25 / 40

  46. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2 for Hegselmann-Krause model τ = 1 26 / 40

  47. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2 for Hegselmann-Krause model τ = 0 . 5 26 / 40

  48. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 2 for Hegselmann-Krause model τ = 0 . 1 26 / 40

  49. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3 for Hegselmann-Krause model ε = 0 . 2 27 / 40

  50. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3 for Hegselmann-Krause model ε = 0 . 05 27 / 40

  51. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Strategy 3 for Hegselmann-Krause model ε = 0 . 01 27 / 40

  52. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Section 2 Sparse Jurdjevic-Quinn stabilization for mean-field equations 28 / 40

  53. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Mean-Field equations Let us consider a large number N of interacting agents: Analysis of a very large-dimensional ODE is hard; Numerical simulations are hard; Control is hard. If agents are all identical, we can replace the positions of agents with the density of the crowd µ . What is the dynamics of µ ? 29 / 40

  54. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Mean-Field equations Let us consider a large number N of interacting agents: Analysis of a very large-dimensional ODE is hard; Numerical simulations are hard; Control is hard. If agents are all identical, we can replace the positions of agents with the density of the crowd µ . What is the dynamics of µ ? 29 / 40

  55. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Mean-Field equations Let us consider a large number N of interacting agents: Analysis of a very large-dimensional ODE is hard; Numerical simulations are hard; Control is hard. If agents are all identical, we can replace the positions of agents with the density of the crowd µ . What is the dynamics of µ ? 29 / 40

  56. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Proposition If f is regular, the mean-field limit of x i = 1 � ˙ f ( x j − x i ) (ODE) N j � = i is ∂ t µ + ∇ · ( F [ µ ] µ ) = 0 . (PDE) where F [ µ ] = f ⋆ µ. Conditions of existence, uniqueness, continuous dependence...? 30 / 40

  57. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Proposition If f is regular, the mean-field limit of x i = 1 � ˙ f ( x j − x i ) (ODE) N j � = i is ∂ t µ + ∇ · ( F [ µ ] µ ) = 0 . (PDE) where F [ µ ] = f ⋆ µ. Conditions of existence, uniqueness, continuous dependence...? 30 / 40

  58. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Proposition Assume that F is defined by F [ µ ] ( x ) := v ( f ⋆ µ ( x )) with v, f Lipschitz. Then there exists a unique solution of the Cauchy problem � ∂ t µ + ∇ · ( F [ µ ] µ ) = 0 µ | t =0 = µ 0 , Remark This is a particular case of a theory in Wasserstein spaces: Ambrosio-Gangbo, Colombo-Garavello-Mercier, Piccoli-Rossi... Proposition (Piccoli-Tosin 2011, Piccoli-Rossi 2013) Numerical schemes are available. 31 / 40

  59. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation N interacting particles x i = 1 ˙ � j f ( x i − x j ) N 32 / 40

  60. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation N interacting particles x i = 1 ˙ � j f ( x i − x j ) N Control of the density 32 / 40

  61. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation N interacting particles Evolution of the density mean-field − − − − − − → x i = 1 ˙ � j f ( x i − x j ) ∂ t µ + ∇ · ( F [ µ ] µ ) = 0 N Control of the density 32 / 40

  62. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation N interacting particles Evolution of the density mean-field − − − − − − → x i = 1 ˙ � j f ( x i − x j ) ∂ t µ + ∇ · ( F [ µ ] µ ) = 0 N ↓ control? Control of the density 32 / 40

  63. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation N interacting particles Evolution of the density mean-field − − − − − − → x i = 1 ˙ � j f ( x i − x j ) ∂ t µ + ∇ · ( F [ µ ] µ ) = 0 N control ↓ ↓ control? Controlled ODE Control of the density x i = 1 � ˙ j f ( x i − x j )+ u i ( t ) N 32 / 40

  64. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation N interacting particles Evolution of the density mean-field − − − − − − → x i = 1 ˙ � j f ( x i − x j ) ∂ t µ + ∇ · ( F [ µ ] µ ) = 0 N control ↓ ↓ control? Controlled ODE mean-field? − − − − − − → Control of the density x i = 1 � ˙ j f ( x i − x j )+ u i ( t ) N 32 / 40

  65. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation How to control the transport equation? How to define controls? Remark Controls on each agent � x i = � ˙ j f ( x j − x i )+ u i ( t ) make no sense in the mean-field limit. Definition The control is a localized vector field χ ω u . It acts as follows ∂ t µ + ∇ · [( F [ µ ] + χ ω u ) µ ] = 0 . We need χ ω u Lipschitz. 33 / 40

  66. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Control of the transport equation How to control the transport equation? How to define controls? Remark Controls on each agent � x i = � ˙ j f ( x j − x i )+ u i ( t ) make no sense in the mean-field limit. Definition The control is a localized vector field χ ω u . It acts as follows ∂ t µ + ∇ · [( F [ µ ] + χ ω u ) µ ] = 0 . We need χ ω u Lipschitz. 33 / 40

  67. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Definition (Sparsity constraint) Fix a small area c > 0 of the configuration space and impose the control set ω ( t ) satisfies | ω ( t ) | ≤ c ; the control function u ( t ) is defined on ω ( t ) with � u ( t ) � ∞ ≤ 1 We aim to generalize Strategy 3 to this setting. Remark One control only, for more controls adapt finite-dimensional. The hardest part is to understand how to generalize objets for the Jurdjevic-Quinn method: proper Lyapunov function, Lie derivatives, steepest descent... 34 / 40

  68. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Definition (Sparsity constraint) Fix a small area c > 0 of the configuration space and impose the control set ω ( t ) satisfies | ω ( t ) | ≤ c ; the control function u ( t ) is defined on ω ( t ) with � u ( t ) � ∞ ≤ 1 We aim to generalize Strategy 3 to this setting. Remark One control only, for more controls adapt finite-dimensional. The hardest part is to understand how to generalize objets for the Jurdjevic-Quinn method: proper Lyapunov function, Lie derivatives, steepest descent... 34 / 40

  69. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations Definition (Sparsity constraint) Fix a small area c > 0 of the configuration space and impose the control set ω ( t ) satisfies | ω ( t ) | ≤ c ; the control function u ( t ) is defined on ω ( t ) with � u ( t ) � ∞ ≤ 1 We aim to generalize Strategy 3 to this setting. Remark One control only, for more controls adapt finite-dimensional. The hardest part is to understand how to generalize objets for the Jurdjevic-Quinn method: proper Lyapunov function, Lie derivatives, steepest descent... 34 / 40

  70. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations The idea is to have Lipschitz-mollified characteristic functions χ η [ a,b ] a − η a b b + η ( a, b, η ) | | ω ( a, b, η ) | ≤ c and η ≥ t − 1 � � Ω t = . At time t n , choose one of the maximizers ( a ∗ , b ∗ , η ∗ ) in Ω t n of V [ e τχ η � [ a,b ] g µ ( t n )] − V [ µ ( t n )] � � � s t n ( a, b, η ) := | L χ η [ a,b ] g V [ µ ( t n )] | = � lim � � � τ � τ → 0 � 35 / 40

  71. Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations If either s t n ( a ∗ , b ∗ , η ∗ ) < t − 1 or Ω t n is empty, then choose the n zero control. a, ¯ η ) ∈ Ω ′ Let the measure evolve up to the existence of (¯ b, ¯ t a, ¯ η ) ≥ 2 t − 1 . such that s t (¯ b, ¯ If s t n ( a ∗ , b ∗ , η ∗ ) ≥ t − 1 n , then choose the control = − χ η ∗ u ( t, · ) [ a ∗ ,b ∗ ] sign( L χ η ∗ [ a ∗ ,b ∗ ] g [ µ ( t )] V [ µ ( t )]) . Let the measure evolve up to one of the following conditions: either s t ( a ∗ , b ∗ , η ∗ ) ≤ t − 1 2 ; a, ¯ or there exists (¯ b, ¯ η ) ∈ Ω ′ t such that a, ¯ s t ( a ∗ , b ∗ , η ∗ ) ≤ (1 − ε ) s t (¯ b, ¯ η ) . Theorem (CPRT, submitted) For each ε ∈ (0 , 1) , this strategy drives the solution to (PDE) to Z = { µ ∈ P c ( R n ) | L f V [ µ ] = L ug V [ µ ] = 0 ∀ u ∈ Lip( R n , R ) } 36 / 40

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