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The role of state constraints for turnpike behaviour and strict dissipativity of optimal control problems Lars Gr une Mathematisches Institut, Universit at Bayreuth based on joint work with Roberto Guglielmi (GSSI, LAquila) Control of

  1. Example 2: a macroeconomic model The second example is a 1d macroeconomic model [Brock/Mirman ’72] Minimise the finite horizon objective with ℓ ( x, u ) = − ln( Ax α − u ) , A = 5 , α = 0 . 34 x + = u with dynamics on X = U = [0 , 10] Here the optimal trajectories are less obvious On infinite horizon, it is optimal to stay at the equilibrium x e ≈ 2 . 2344 with ℓ ( x e , u e ) ≈ 1 . 4673 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 8

  2. Example 2: a macroeconomic model The second example is a 1d macroeconomic model [Brock/Mirman ’72] Minimise the finite horizon objective with ℓ ( x, u ) = − ln( Ax α − u ) , A = 5 , α = 0 . 34 x + = u with dynamics on X = U = [0 , 10] Here the optimal trajectories are less obvious On infinite horizon, it is optimal to stay at the equilibrium x e ≈ 2 . 2344 with ℓ ( x e , u e ) ≈ 1 . 4673 One may thus expect that finite horizon optimal trajectories also stay for a long time near that equilibrium Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 8

  3. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectory for N = 5 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  4. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 7 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  5. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 9 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  6. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 11 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  7. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 13 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  8. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 15 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  9. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 17 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  10. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 19 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  11. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 21 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  12. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 23 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  13. Example 2: optimal trajectories 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Optimal trajectories for N = 5 , . . . , 25 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

  14. How to formalize the turnpike property? 1.8 1.6 1.4 1.2 1 x(n) 0.8 0.6 0.4 0.2 0 −0.2 0 5 10 15 20 25 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 10

  15. How to formalize the turnpike property? 1.8 1.6 1.4 1.2 1 x(n) 0.8 0.6 0.4 0.2 0 -0.2 0 5 10 15 20 25 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 10

  16. How to formalize the turnpike property? 1.8 1.6 1.4 1.2 1 x(n) 0.8 0.6 0.4 0.2 0 -0.2 0 5 10 15 20 25 n Number of points outside the blue neighbourhood is bounded by a number independent of N (here: by 8) Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 10

  17. The turnpike property: formal definitions Let x e be an equilibrium, i.e., f ( x e , u e ) = x e Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 11

  18. The turnpike property: formal definitions Let x e be an equilibrium, i.e., f ( x e , u e ) = x e Turnpike property: For each ε > 0 and ρ > 0 there is C ρ,ε > 0 such that for all N ∈ N all optimal trajectories x ⋆ starting in B ρ ( x e ) satisfy the inequality � � � � � x ⋆ ( k ) − x e � ≥ ε # k ∈ { 0 , . . . , N − 1 } ≤ C ρ,ε � Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 11

  19. The turnpike property: formal definitions Let x e be an equilibrium, i.e., f ( x e , u e ) = x e Turnpike property: For each ε > 0 and ρ > 0 there is C ρ,ε > 0 such that for all N ∈ N all optimal trajectories x ⋆ starting in B ρ ( x e ) satisfy the inequality � � � � � x ⋆ ( k ) − x e � ≥ ε # k ∈ { 0 , . . . , N − 1 } ≤ C ρ,ε � Near equilibrium turnpike property: For each ε > 0 , δ > 0 and ρ > 0 there is C ρ,ε,δ > 0 such that for all x ∈ X and N ∈ N , all trajectories x u with x u (0) = x ∈ B ρ ( x e ) and J N ( x, u ) ≤ Nℓ ( x e , u e ) + δ satisfy the inequality � � � � � x u ( k ) − x e � ≥ ε # k ∈ { 0 , . . . , N − 1 } ≤ C ρ,ε,δ � Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 11

  20. History Apparently first described by [von Neumann 1945] Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

  21. History Apparently first described by [von Neumann 1945] Name “turnpike property” coined by [Dorfman/Samuelson/Solow 1957] Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

  22. History Apparently first described by [von Neumann 1945] Name “turnpike property” coined by [Dorfman/Samuelson/Solow 1957] 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

  23. History Apparently first described by [von Neumann 1945] Name “turnpike property” coined by [Dorfman/Samuelson/Solow 1957] Extensively studied in the 1970s in mathematical economy, cf. survey [McKenzie 1983] Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

  24. History Apparently first described by [von Neumann 1945] Name “turnpike property” coined by [Dorfman/Samuelson/Solow 1957] Extensively studied in the 1970s in mathematical economy, cf. survey [McKenzie 1983] Renewed interest in recent years [Zaslavski ’14, Tr´ elat/Zuazua ’15, Faulwasser et al. ’15, . . . ] Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

  25. History Apparently first described by [von Neumann 1945] Name “turnpike property” coined by [Dorfman/Samuelson/Solow 1957] Extensively studied in the 1970s in mathematical economy, cf. survey [McKenzie 1983] Renewed interest in recent years [Zaslavski ’14, Tr´ elat/Zuazua ’15, Faulwasser et al. ’15, . . . ] Many applications, e.g., structural insight in economic equilibria; synthesis of optimal trajectories [Anderson/Kokotovic ’87] Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

  26. Application: Model predictive control Turnpike properties are also pivotal for analysing economic Model Predictive Control (MPC) schemes Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 13

  27. Application: Model predictive control Turnpike properties are also pivotal for analysing economic Model Predictive Control (MPC) schemes MPC is a method in which an optimal control problem on an infinite horizon ∞ � minimise J ∞ ( x, u ) = ℓ ( x u ( n ) , u ( n )) u n =0 is approximated by the iterative solution of finite horizon problems N − 1 � minimise J N ( x, u ) = ℓ ( x u ( k ) , u ( k )) u n =0 with fixed N ∈ N Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 13

  28. MPC from the trajectory point of view Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  29. MPC from the trajectory point of view x x 0 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  30. MPC from the trajectory point of view x x 1 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  31. MPC from the trajectory point of view x x 1 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  32. MPC from the trajectory point of view x x 2 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  33. MPC from the trajectory point of view x x 2 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  34. MPC from the trajectory point of view x x 3 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  35. MPC from the trajectory point of view x ... x 3 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  36. MPC from the trajectory point of view x ... x 4 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  37. MPC from the trajectory point of view x ... ... x 4 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  38. MPC from the trajectory point of view x ... ... x 5 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  39. MPC from the trajectory point of view x ... ... ... x 5 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  40. MPC from the trajectory point of view x ... ... ... x 6 n 0 1 2 3 4 5 6 black = predictions (open loop optimization) red = MPC closed loop Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

  41. Approximation result for MPC If the finite horizon problems have the turnpike property, then a rigorous approximation result can be proved Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 15

  42. Approximation result for MPC If the finite horizon problems have the turnpike property, then a rigorous approximation result can be proved The result exploits that the red closed loop trajectory approximately follows the first part of the black predictions up to the equilibrium Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 15

  43. Approximation result for MPC If the finite horizon problems have the turnpike property, then a rigorous approximation result can be proved The result exploits that the red closed loop trajectory approximately follows the first part of the black predictions up to the equilibrium We illustrate this behaviour by our second example for N = 10 Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 15

  44. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  45. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  46. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  47. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  48. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  49. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  50. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  51. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  52. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  53. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  54. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  55. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  56. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  57. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  58. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  59. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  60. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  61. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  62. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  63. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  64. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  65. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  66. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  67. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  68. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  69. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  70. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  71. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

  72. MPC for Example 2 5.5 5 4.5 4 3.5 3 x(n) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 n Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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