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Sensitivity and turnpike results for the optimal control of PDEs and their use for model predictive control Lars Gr une Mathematisches Institut, Universit at Bayreuth based on joint work with Manuel Schaller, Anton Schiela (both

  1. Why use MPC? What is the advantage of MPC over other methods of solving optimal control problems? significantly reduced computational complexity � real time capability ability to react to perturbations applicability to real-world industrial applications applicability to problems in which data becomes available online But: The trajectory delivered by MPC can be far from optimal! Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

  2. Why use MPC? What is the advantage of MPC over other methods of solving optimal control problems? significantly reduced computational complexity � real time capability ability to react to perturbations applicability to real-world industrial applications applicability to problems in which data becomes available online But: The trajectory delivered by MPC can be far from optimal! � Key questions in this talk: When does MPC yield closed loop trajectories with approximately optimal performance? Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

  3. Why use MPC? What is the advantage of MPC over other methods of solving optimal control problems? significantly reduced computational complexity � real time capability ability to react to perturbations applicability to real-world industrial applications applicability to problems in which data becomes available online But: The trajectory delivered by MPC can be far from optimal! � Key questions in this talk: When does MPC yield closed loop trajectories with approximately optimal performance? How can we implement MPC efficiently for PDEs? Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

  4. What makes model predictive control work?

  5. The turnpike property The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time [von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83] 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

  6. The turnpike property The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time [von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83] In the simplest case, this particular trajectory is an equilibrium (but extensions to periodic orbits or more general time varying trajectories exist) Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

  7. The turnpike property The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time [von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83] In the simplest case, this particular trajectory is an equilibrium (but extensions to periodic orbits or more general time varying trajectories exist) In this talk we stick to the equilibrium setting Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

  8. The turnpike property The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time [von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83] In the simplest case, this particular trajectory is an equilibrium (but extensions to periodic orbits or more general time varying trajectories exist) In this talk we stick to the equilibrium setting We illustrate it by a simple discrete time example Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

  9. Example: a macroeconomic model Consider a classical 1d macroeconomic model [Brock/Mirman ’72] Minimize the finite horizon objective � N − 1 k =0 ℓ ( y ( k ) , u ( k )) with ℓ ( y, u ) = − ln( Ay α − u ) , A = 5 , α = 0 . 34 and dynamics y ( k + 1) = u ( k ) on Y = U = [0 , 10] Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

  10. Example: a macroeconomic model Consider a classical 1d macroeconomic model [Brock/Mirman ’72] Minimize the finite horizon objective � N − 1 k =0 ℓ ( y ( k ) , u ( k )) with ℓ ( y, u ) = − ln( Ay α − u ) , A = 5 , α = 0 . 34 and dynamics y ( k + 1) = u ( k ) on Y = U = [0 , 10] y = invested capital; u = investment in next time step Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

  11. Example: a macroeconomic model Consider a classical 1d macroeconomic model [Brock/Mirman ’72] Minimize the finite horizon objective � N − 1 k =0 ℓ ( y ( k ) , u ( k )) with ℓ ( y, u ) = − ln( Ay α − u ) , A = 5 , α = 0 . 34 and dynamics y ( k + 1) = u ( k ) on Y = U = [0 , 10] y = invested capital; u = investment in next time step Ay α = capital after one time step Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

  12. Example: a macroeconomic model Consider a classical 1d macroeconomic model [Brock/Mirman ’72] Minimize the finite horizon objective � N − 1 k =0 ℓ ( y ( k ) , u ( k )) with ℓ ( y, u ) = − ln( Ay α − u ) , A = 5 , α = 0 . 34 and dynamics y ( k + 1) = u ( k ) on Y = U = [0 , 10] y = invested capital; u = investment in next time step Ay α = capital after one time step Ay α − u = consumed capital; ln( · ) = utility function Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

  13. Example: a macroeconomic model Consider a classical 1d macroeconomic model [Brock/Mirman ’72] Minimize the finite horizon objective � N − 1 k =0 ℓ ( y ( k ) , u ( k )) with ℓ ( y, u ) = − ln( Ay α − u ) , A = 5 , α = 0 . 34 and dynamics y ( k + 1) = u ( k ) on Y = U = [0 , 10] y = invested capital; u = investment in next time step Ay α = capital after one time step Ay α − u = consumed capital; ln( · ) = utility function On infinite horizon, it is optimal to stay at the equilibrium y e ≈ 2 . 2344 with ℓ ( y e , u e ) ≈ 1 . 4673 Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

  14. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  15. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  16. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  17. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  18. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  19. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  20. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  21. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  22. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  23. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  24. Illustration of the Turnpike Property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  25. Illustration of the Turnpike Property 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 The turnpike property makes MPC work... Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

  26. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  27. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  28. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  29. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  30. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  31. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  32. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  33. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  34. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  35. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  36. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  37. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  38. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  39. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  40. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  41. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  42. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  43. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  44. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  45. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  46. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  47. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  48. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  49. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  50. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  51. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  52. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  53. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  54. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  55. MPC and the turnpike property 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, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

  56. Approximation properties of MPC In order to formalize how good MPC approximates the infinite horizon problem, we define y MPC ( t, y 0 ) = solution generated by MPC starting in y 0 u MPC ( t ) = control function generated by MPC � S J MPC ( y 0 ) = ℓ ( y MPC ( t, y 0 ) , u MPC ( t )) dt S 0 Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 13/40

  57. Approximation properties of MPC In order to formalize how good MPC approximates the infinite horizon problem, we define y MPC ( t, y 0 ) = solution generated by MPC starting in y 0 u MPC ( t ) = control function generated by MPC � S J MPC ( y 0 ) = ℓ ( y MPC ( t, y 0 ) , u MPC ( t )) dt S 0 Furthermore, we define � � � continuous , β ( r, t ) ր ∞ as r → ∞ � β : R 2 KL := ≥ 0 → R ≥ 0 � β (0 , t ) = 0 , β ( r, t ) ց 0 as t → ∞ � β( ) r*, t β( ) r, t* r (0, 0) t (0, 0) Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 13/40

  58. Approximation properties of MPC Theorem: If the turnpike property at an optimal equilibrium ( y e , u e ) and suitable controllability and regularity conditions hold, then there exist ε 1 ( T ) , ε 2 ( S ) → 0 as T → ∞ and S → ∞ , such that the following properties hold Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

  59. Approximation properties of MPC Theorem: If the turnpike property at an optimal equilibrium ( y e , u e ) and suitable controllability and regularity conditions hold, then there exist ε 1 ( T ) , ε 2 ( S ) → 0 as T → ∞ and S → ∞ , such that the following properties hold (1) Approximate average optimality: 1 S J MPC ( y 0 ) ≤ ℓ ( y e , u e ) + ε 1 ( T ) lim sup S S →∞ Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

  60. Approximation properties of MPC Theorem: If the turnpike property at an optimal equilibrium ( y e , u e ) and suitable controllability and regularity conditions hold, then there exist ε 1 ( T ) , ε 2 ( S ) → 0 as T → ∞ and S → ∞ , such that the following properties hold (1) Approximate average optimality: 1 S J MPC ( y 0 ) ≤ ℓ ( y e , u e ) + ε 1 ( T ) lim sup S S →∞ (2) Practical asymptotic stability: there is β ∈ KL : � y MPC ( t, y 0 ) − y e � ≤ β ( � y 0 − y e � , t ) + ε 1 ( T ) for all k ∈ N Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

  61. Approximation properties of MPC Theorem: If the turnpike property at an optimal equilibrium ( y e , u e ) and suitable controllability and regularity conditions hold, then there exist ε 1 ( T ) , ε 2 ( S ) → 0 as T → ∞ and S → ∞ , such that the following properties hold (1) Approximate average optimality: 1 S J MPC ( y 0 ) ≤ ℓ ( y e , u e ) + ε 1 ( T ) lim sup S S →∞ (2) Practical asymptotic stability: there is β ∈ KL : � y MPC ( t, y 0 ) − y e � ≤ β ( � y 0 − y e � , t ) + ε 1 ( T ) for all k ∈ N (3) Approximate transient optimality: for all S ∈ N : J MPC ( y 0 ) ≤ J S ( y 0 , u ) + Sε 1 ( T ) + ε 2 ( S ) S for all admissible u with � y ( S, y 0 , u ) − y e � ≤ β ( � y 0 − y e � , S ) + ε 1 ( T ) Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

  62. Illustration of (2) and (3) y e y t Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

  63. Illustration of (2) and (3) y ε (T) 1 e y t Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

  64. Illustration of (2) and (3) y ε (T) 1 e y t (2): y MPC ( t ) converges to the ε 1 ( T ) -ball around y e Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

  65. Illustration of (2) and (3) y ε (T) 1 e y S t (2): y MPC ( t ) converges to the ε 1 ( T ) -ball around y e Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

  66. Illustration of (2) and (3) y ε (T) 1 e y S t (2): y MPC ( t ) converges to the ε 1 ( T ) -ball around y e Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

  67. Illustration of (2) and (3) y ε (T) 1 e y S t (2): y MPC ( t ) converges to the ε 1 ( T ) -ball around y e Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

  68. Illustration of (2) and (3) y ε (T) 1 e y S t (2): y MPC ( t ) converges to the ε 1 ( T ) -ball around y e Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

  69. Illustration of (2) and (3) y ε (T) 1 e y S t (2): y MPC ( t ) converges to the ε 1 ( T ) -ball around y e (3): cost of all other trajectories reaching the ball at time K is (3): higher than that of y MPC ( t ) up to the error Sε 1 ( T ) + ε 2 ( S ) Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

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