The HJB-POD approach for infinite dimensional control problems

Transcription

1 The HJB-POD approach for infinite dimensional control problems M. Falcone works in collaboration with A. Alla, D. Kalise and S. Volkwein Università di Roma La Sapienza OCERTO Workshop Cortona, June 22, 26 M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach / 48

2 Outline Outline HJ equations, DP schemes and feedback synthesis HJ Equations Numerical scheme for HJ equation 2 Efficient numerical methods for HJ equations A quick overview Accelerated iterative schemes 3 HJB-POD method for high dimensional problem A-priori estimates for the HJB approximation A-priori estimates for the HJB-POD approximation 4 Numerical Tests M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 2 / 48

3 Outline Introduction The Dynamic Programming Principle allows to derive a first order partial differential equation describing the value function associated to the optimal control problem (in finite or infinite dimension). The theory of viscosity solutions allows to characterize the value function as the unique weak solution of the Bellman equation. This characterization has been used also to construct numerical schemes for the value function and to compute optimal feedbacks. Reference M. Bardi, I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, 997. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 3 / 48

4 Outline DP s advantages and disadvantages PROS. The characterization of the value function is valid for all classical problems in any dimension. 2. The approximation is based on a-priori error estimates in L and is valid in any dimension 3. DP (semi-lagrangian) schemes can work on structured and unstructured grids. 4. The computation of feedbacks is almost built in and there are nice results in low dimension. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 4 / 48

5 Outline DP s advantages and disadvantages PROS. The characterization of the value function is valid for all classical problems in any dimension. 2. The approximation is based on a-priori error estimates in L and is valid in any dimension 3. DP (semi-lagrangian) schemes can work on structured and unstructured grids. 4. The computation of feedbacks is almost built in and there are nice results in low dimension. CONS The "curse of dimensionality" makes the problem difficult to solve in high dimension due to. computational cost 2. huge memory allocations. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 4 / 48

6 Outline HJ equations, DP schemes and feedback synthesis HJ equations, DP schemes and feedback synthesis HJ Equations Numerical scheme for HJ equation 2 Efficient numerical methods for HJ equations A quick overview Accelerated iterative schemes 3 HJB-POD method for high dimensional problem A-priori estimates for the HJB approximation A-priori estimates for the HJB-POD approximation 4 Numerical Tests M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 5 / 48

7 HJ equations, DP schemes and feedback synthesis HJ Equations HJB equation for the infinite horizon problem Controlled Dynamics and Cost Functional { ẏ(t) = f (y(t), u(t)), t (t, + ] y(t ) = x, Infinite horizon cost functional Value Function J x (y, u) = + v(x) := g(y(s), u(s))e λs ds inf J x(y, u( )). u( ) U M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 6 / 48

8 HJ equations, DP schemes and feedback synthesis Numerical scheme for HJ equation Value and HJB equation: infinite horizon problem Dynamic Programming Principle { τ } v(x) = min e λs g(y x (s), u(s)) ds + v(y x (τ))e λτ u U t By Dynamic Programming we get the stationary Bellman equation λv(x) + max{ f (x, u) v(x) g(x, u)} =, x u U Rn Since the value function is in general not regular we need to use weak solutions, typically Lipschitz continuous. The value function is the unique viscosity solution of the Bellman equation. The construction of the approximation scheme can be obtained via a discrete dynamic programming approach. State constraints can also be included. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 7 / 48

9 HJ equations, DP schemes and feedback synthesis Synthesis of feedback controls Numerical scheme for HJ equation The numerical solution of optimal control problems via HJB PDEs leads to the computation of feedback controls for generic nonlinear Lipschitz continuous vectorfields and costs. Solving λv(x) + max{ f (x, u) v(x) g(x, u)} =, x u U Ω we get the value function on a grid and we extend it to the whole domain. Then, we can also compute a feedback map u : Ω U u (x) arg min{f (x, u) v(x) + g(x, u)}, x u U Ω which is used to compute optimal trajectories by an ODE scheme. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 8 / 48

10 HJ equations, DP schemes and feedback synthesis Numerical scheme for HJ equation The Zermelo navigation problem Value Function Feedback M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 9 / 48

11 Outline Efficient numerical methods for HJ equations HJ equations, DP schemes and feedback synthesis HJ Equations Numerical scheme for HJ equation 2 Efficient numerical methods for HJ equations A quick overview Accelerated iterative schemes 3 HJB-POD method for high dimensional problem A-priori estimates for the HJB approximation A-priori estimates for the HJB-POD approximation 4 Numerical Tests M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach / 48

12 Efficient numerical methods for HJ equations A quick overview How can we compute the value function? The bottleneck of the DP approach is the computation of the value function, since this requires to solve a non linear PDE in high-dimension. This is a challenging problem due to the huge number of nodes involved and to the singularities of the solution. This goal has motivated new efforts in several directions: Domain Decomposition (DD) Fast Marching Methods (FMM) Accelerated iterative schemes M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach / 48

13 Efficient numerical methods for HJ equations Accelerated iterative schemes Semi-Lagrangian discretization of HJB Dynamic Programming Principle { τ } v(x) = min e λs g(y x (s), u(s)) ds + v(y x (τ))e λτ u U t Time-Discrete Approximation via Value Iteration V k+ i = min u U {e λ t V k (x i + t f (x i, u)) + t g (x i, u)} Fix a grid in Ω with Ω R n bounded, Steps x. Nodes: {x,..., x N }, Discrete solution: V i v(x i ). Stability for large time steps t M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 2 / 48

14 Efficient numerical methods for HJ equations Accelerated iterative schemes Semi-Lagrangian discretization of HJB The most standard way to solve this system is the Value Iteration (VI). Fully discrete SL-FEM/ Value Iteration (VI) scheme ( T (V k ) Some advantages: V k+ ) = T (V k ), for i =,..., N i min u U {e λ t I[V k ](x i + t f (x i, u)) + g(x i, u)} Simple to implement (for I = I,the P interpolation operator) (VI) Converges under very general assumptions. WARNING: Rather expensive in terms of CPU time, since β = e λ t the Lipschitz constant of T goes to when t. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 3 / 48

15 Efficient numerical methods for HJ equations Accelerated iterative schemes Semi-Lagrangian discretization of HJB Fully-Discrete Approximation (Value Iteration) V k+ i = min u U {e λ t I [ V k] (x i + t f (x i, u)) + t L (x i, u)} This algorithm converges for any initial guess V. Error Estimate: [F. 987] max v(x i ) V i C t /2 L + f x i N G λ(λ L f ) t. N G = number of nodes, L f = Lipschitz constant of the dynamics f. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 4 / 48

16 Efficient numerical methods for HJ equations Accelerated iterative schemes Policy iteration An alternative form to solve this problem is the iteration in the policy space [Bellman 955, Howard 96]. Fully discrete SL-FEM/ Policy Iteration (PI) scheme Fix u i U, for i =,..., K. 2 Solve (V k ) i = βi [V k ](x i + t f (x i, u k i )) + t g(x i, u k i ). 3 Update u k+ i = argmin{i [V k ](x i + t f (x i, u)) + t g(x i, u)}. u U 4 Repeat until matching convergence criteria (can be set on V or u).typically we use V k+ V k < ɛ as stopping rule. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 5 / 48

17 Efficient numerical methods for HJ equations Numerical Fact #: PI is faster Accelerated iterative schemes V k V evolution for VI (2D MTP) 2 2 DoF 4 2 DoF 8 2 DoF 6 2 DoF L error # of iterations M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 6 / 48

18 Efficient numerical methods for HJ equations Numerical Fact #: PI is faster Accelerated iterative schemes V k V evolution for PI (2D MTP) 2 2 DoF 4 2 DoF 8 2 DoF 6 2 DoF L error # of iterations M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 7 / 48

19 Efficient numerical methods for HJ equations Accelerated iterative schemes Numerical Fact #2: PI is sensitive to u i. V k V evolution for PI with different u i.9.8 Guess Guess 2 Guess 3 Guess 4 L error # of iterations M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 8 / 48

20 Efficient numerical methods for HJ equations Accelerated iterative schemes Numerical Fact #2: PI is sensitive to u i. PI inner solver iteration subcount (2D MTP) 8 7 Guess Guess 4 Guess 3 6 Subiterations count # of global iterations M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 9 / 48

21 Efficient numerical methods for HJ equations Accelerated iterative schemes Num. Fact #3: VI is fast in coarse meshes. V k V evolution for VI (2D MTP) 2 2 DoF 4 2 DoF 8 2 DoF 6 2 DoF L error # of iterations M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 2 / 48

22 Efficient numerical methods for HJ equations Accelerated iterative schemes The accelerated algorithm (Alla, F., Kalise, 25) Theoretical results have established a link between VI and PI in control problems and games [Puterman and Brumelle (979), Bokanowski, Maroso and Zidani (29)]. Numerically, PI strongly depends on a good initialization. Similar approaches have been studied by González and Sagastizábal (99), Chow (99), Seeck (997), and Grüne (24). The Accelerated PI (API) algorithm Consider two mesh parameters k and k 2, k > k 2. Set V k. 2 Perform a coarse VI in the k mesh with result V k. 3 Vk 2 = I [Vk ] and uk 2 = argmin{i [Vk 2 ](x i + hf (x i, u))}. u U 4 start PI solver over the k 2 -mesh. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 2 / 48

23 Efficient numerical methods for HJ equations Test : different performances Accelerated iterative schemes L -norm V k V evolution. 2D EIKONAL EQUATION,6 CONTROLS, DELTAX=.25. Value Iteration Policy Iteration Accelerated PI L inf NORM CPU TIME M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 22 / 48

24 Outline HJB-POD method for high dimensional problem HJ equations, DP schemes and feedback synthesis HJ Equations Numerical scheme for HJ equation 2 Efficient numerical methods for HJ equations A quick overview Accelerated iterative schemes 3 HJB-POD method for high dimensional problem A-priori estimates for the HJB approximation A-priori estimates for the HJB-POD approximation 4 Numerical Tests M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 23 / 48

25 HJB-POD method for high dimensional problem Control of Partial Differential Equations via DP The discretization of a PDE leads to a large system of ODEs. The approximation of the correspondent HJB equations becomes unfeasible because of the curse of dimensionality. Model Reduction via Proper Ortoghonal Decomposition POD decomposition allows to reduce the number of variables to approximate partial differential equations. GOAL: to approximate PDE optimal control problems in a rather small dimension via POD using numerical schemes for HJB equations. References Kunisch and Volkwein (2,...), Kunisch, Volkwein and Xie (24), Alla-F. (23, 24), Alla-Hinze (25). M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 24 / 48

26 HJB-POD method for high dimensional problem Proper Orthogonal Decomposition and SVD Given snapshots : (y(t, u),..., y(t m, u)) R n We look for an orthonormal basis {ψ i } l i= in Rm with l min{n, m} s.t. J(ψ,..., ψ l ) = m l 2 d α j y j y j, ψ i ψ i = j= i= i=l+ σ 2 i reaches a minimum where {α j } n j= R+. min J(ψ,..., ψ l ) s.t. ψ i, ψ j = δ ij Singular Value Decomposition: Y = ΨΣV T. For l {,..., d = rank(y )}, {ψ i } l i= is called the POD basis of rank l. Ansatz: l y(x, t) yi l (t)ψ i(x) i= M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 25 / 48

27 HJB-POD method for high dimensional problem Reduced Order Modeling Control Problem min J l (y l, u) y l satisfies the reduced dynamics { ẏ l (t) = F l (y l (t), u(t)) t >, y l (t ) = y l Rl. The cost functional is: J l y l (y l, u) = Reduced Value Function v(y l ) = g l (y l (t), u(t))e λt dt inf J l (y l, u) u U y l ad M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 26 / 48

28 HJB-POD method for high dimensional problem Reduced Domain (in the POD space) for HJB We are going to solve the HJB equation in a box: [a, b ]... [a l, b l ] The box must be chosen such that the dynamic contains all the possible controlled trajectories. We discretize the control space U by a discrete set {u,..., u M }. l y(t) = < y(t, u j ), ψ i > ψ i = i= and we choose the box to guarantee l i= y l i (t, u j)ψ i y l i (t, u j) [a i, b i ], j =,..., M. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 27 / 48

29 HJB-POD method for high dimensional problem Back to the infinite horizon problem A-priori estimates for the HJB approximation Let us consider the controlled dynamics ẏ(t) = f ( y(t), u(t) ) R n for t >, y() = y R n () where f : R n R m R n The infinite horizon cost functional is J(y, u) = where λ > and g : R n R m R. The set of admissible controls has the form g ( y(t), u(t) ) e λt dt (2) U ad = { u U u(t) U ad for almost all t }, where U = L 2 (, ; R m ) and U ad R m is a compact, convex subset. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 28 / 48

30 HJB-POD method for high dimensional problem A-priori estimates for the HJB approximation Variational formulation of the dynamics Let M R n n denote a symmetric, positive definite (mass) matrix with smallest and largest positive eigenvalues λ min and λ max, respectively. We introduce the following weighted inner product in R n : y, ỹ M = y Mỹ for y, ỹ R n, By M =, /2 M we define the associated induced norm. Recall that we have Then, y is a trajectory if λ min y 2 2 y 2 M λ max y 2 2 for all y R n. ẏ(t) f (y(t), u(t)), ϕ M = for all ϕ R n and for almost all t >, y() y, ϕ M = for all ϕ R n (3) M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 29 / 48

31 HJB-POD method for high dimensional problem The infinite horizon problem A-priori estimates for the HJB approximation Assume that the dynamics has a unique solution y = y(u; y ) Y = H (, ; R n ) for every admissible control u U ad and for every initial condition y R n, denoted by y(u; y ). Reduced cost functional Ĵ(u; y ) = J(y(u; y ), u) for u U ad and y R n, Then, our optimal control problem for y R n is min Ĵ(u; y ). u U ad ( P) M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 3 / 48

32 HJB-POD method for high dimensional problem ASSUMPTIONS A-priori estimates for the HJB approximation (A) f : R n R m R n is continuous and f (y, u) f (ỹ, u) 2 L f y ỹ 2 for all y, ỹ R n and u U ad Moreover, f (y, u) M f for all (y, u) Ω U ad. (A2) g : R n R m R n is continuous and globally Lipschitz continuous. Moreover, g(y, u) M g for all (y, u) Ω U ad. If (A) (A2) hold and λ > L f, v h is globally Lipschitz-continuous v h (y) v h (ỹ) L g λ L f y ỹ 2 y, ỹ Ω and h [, /λ) M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 3 / 48

33 HJB-POD method for high dimensional problem A-priori estimates for the HJB approximation A-priori estimate for HJB [F87] Theorem Let (A) (A2) hold and λ > max{l g, L f }. Let v and v h be the continuous and semi-discrete solutions respectively. Moreover, assume semiconcavity, i.e. f (y + ỹ, u) 2f (y, u) + f (y ỹ, u) 2 C f ỹ 2 2, g(y + ỹ, u) 2g(y, u) + g(y ỹ, u) C g ỹ 2 2 (4) for all (y, ỹ, u) R n R n U ad. Then, sup y R n v(y) v h (y) Ch for any h [, /λ). M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 32 / 48

34 HJB-POD method for high dimensional problem A-priori estimates for the HJB-POD approximation An estimate for the HJB-POD approximation We introduce two different POD approximations for the HJB equation. A first estimate is based on the fully discrete HJB equation, where we project all vertices {y i } n S i= into Rl by setting y l i = Ψ My i for i =,..., n S. Here we assume yi l yj l for i, j {,..., n S } with i j. Then, a POD discretization of HJB in the reduced space is given by v l hk (y l i ) = min u U ad { ( λh)v l hk ( y l i + hf l (y l i, u)) + hg l (y l i, u)} (5) for i n S. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 33 / 48

35 HJB-POD method for high dimensional problem A-priori estimates for the HJB-POD approximation An estimate for the HJB-POD approximation Let us define the mapping ṽ l hk : Ω R by ṽ l hk (y) = v l hk (Ψ My) for all y Ω with Ψ My Ω. Thus, (5) can be written as ṽhk l (y { i) = min ( λh)ṽ l ( hk yi + hf (P l y i, u) ) + hg(p l y i, u) } (6) u U ad for i n S. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 34 / 48

36 HJB-POD method for high dimensional problem A-priori estimates for the HJB-POD approximation An estimate for the HJB-POD approximation To simplify, let us assume the invariance condition for the reduced order dynamics y i + hf (P l y i, u) Ω for i =,..., n S, u U ad (A3) Theorem Assume that (A) (A2), (A3) and λ > L f hold. Then, there exist two constants Ĉ, Ĉ such that ( sup v h (y) ṽhk l (y) k Ĉ y Ω h + ns ) /2 Ĉ y i P l y i 2 2 i= for any h [, /λ) M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 35 / 48

37 HJB-POD method for high dimensional problem A-priori estimates for the HJB-POD approximation An estimate for the HJB-POD approximation By the previous results we conclude Theorem Assume that (A) (A2), (A3) and (A4) hold. Let f, g satisfy the semiconcavity conditions. If λ > max{l f, L g }, then there exists constants C, C, C 2 such that ( sup v(y) ṽhk l (y) k C h + C y Ω h + C ns ) /2 2 y i P l y i 2 2 i= for any h [, /λ). M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 36 / 48

38 Outline Numerical Tests HJ equations, DP schemes and feedback synthesis HJ Equations Numerical scheme for HJ equation 2 Efficient numerical methods for HJ equations A quick overview Accelerated iterative schemes 3 HJB-POD method for high dimensional problem A-priori estimates for the HJB approximation A-priori estimates for the HJB-POD approximation 4 Numerical Tests M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 37 / 48

39 Numerical Tests Test 3: Advection-Diffusion Equation Advection-Diffusion Equation y t εy xx + βy x = χ Ωc (x)u(t) in Ω (, ], y(, ) = y, in Ω, y(, t) = in Ω (, T ), (7) Cost Functional Parameters J(y, u, t) = ( y(x, τ) 2 + γ u(τ) 2) e λτ dτ. ε =., β =, γ =., y (x) =.5 sin(πx), Ω = [, 2], Ω c = (.5, ), U = [ 2.2, ]; Snapshots: x =., t =.. VF: x = {.,.5}, t =. x; Trajectories: t =.; M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 38 / 48

40 Numerical Tests Test 3: Advection-Diffusion Equation time time time Controls time time Figure: Test 3: Uncontrolled (left), LQR optimal (middle), HJB-POD (right). M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 39 / 48

41 Numerical Tests Test 3: Advection-Diffusion Equation K=. K= k=. k= K=. K= Number of POD basis Number of POD basis Number of POD basis Figure: Evaluation of the cost functional (left), L 2 error for y(u l ) and y l (u l ) (middle) and L 2 -error between LQR solution and y(u l ). M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 4 / 48

42 Numerical Tests Test 4: Semi-linear Equation Semi-Linear Equation y t εy xx + βy x + µ(y y 3 ) = y (x)u(t) in Ω (, ], y(, ) = y, in Ω, y(, t) = in Ω (, T ), Cost Functional Parameters J(y, u, t) = ( y(x, τ) 2 + γ u(τ) 2) e λτ dτ. ε =. = β, µ =, γ =., y (x) = sin(πx), Ω = [, ], U = [, ]; Snapshots: x =., t =.. VF: x = {.,.5}, t =. x; Trajectories: t =.; M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 4 / 48

43 Numerical Tests Test 4: Semi-linear Equation time time time time time Figure: Uncontrolled (left), Optimal (middle) and optimal HJB control (right). M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 42 / 48

44 Test 5: Wave Equation Numerical Tests Damped Wave Equation y tt y xx βy xxt = χ Ωc (x)u(t) in Ω (, ], y(, ) = y, y t (, ) = y in Ω, y(, t) = in Ω (, T ), Cost Functional Parameters J(y, u, t) = ( y(x, τ) 2 + γ u(τ) 2) e λτ dτ. β =.5, γ =., y (x) = sin(πx), y (x) =, Ω = [, ], Ω c = (.4,.6), U = [.2,.6]; Snapshots: x =., t =.. VF: t =., x =.; Trajectories: t =.; M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 43 / 48

45 Numerical Tests Test 5: Wave Equation, Optimal Solution TIME TIME `=3 `=4 `=5.2 TIME k =. k = Table: H error between LQR control and HJB-POD approx. at time t = 4. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 44 / 48

46 Numerical Tests Test 5: Wave Equation, Stabilization of the feedback Figure: Feedback Control with Chattering (left) LQR control (right)), M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 45 / 48

47 Numerical Tests Wave Equation: Stabilization of the feedback Figure: Feedback Control with Chattering (left), Stabilized feedback control with limited chattering (right) M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 46 / 48

48 Numerical Tests CONCLUSIONS Accelerated Policy Iteration speeds up the numerical approximation of the value function Model reduction via POD is crucial in order to make the problem feasible Feedback Control for PDEs is obtained via the POD-HJB approach The chattering of the feedback control for finite and infinite dimensional problem can be reduced A-priori error estimation are available M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 47 / 48

49 Numerical Tests References for this talk A. Alla, M. Falcone An adaptive POD approximation method for the control of advection-diffusions equation, 23. A. Alla, M. Falcone, D. Kalise, An efficient Policy Iteration algorithm for Dynamic Programming equations, 25. A. Alla, M. Falcone, D. Kalise, A HJB-POD feedback synthesis approach for the wave equation, 25. A. Alla, M. Falcone, S. Volkwein, Error Analysis for POD approximations of infinite horizon problems via the dynamic programming principle, submitted 25. M. Bardi, I. Capuzzo Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhauser,Basel, 997. M. Falcone, R. Ferretti, Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations, SIAM, 24. K. Kunisch, S. Volkwein, L. Xie, HJB-POD Based Feedback Design for the Optimal Control of Evolution Problems, 24. S. Volkwein. Model Reduction using Proper Orthogonal Decomposition, Lecture Notes, 23. M. Falcone (Università di Roma La Sapienza ) The HJB-POD approach 48 / 48