Quick reachability and proper extension for problems with unbounded controls
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(2) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. .. .. Quick reachability and proper extension for problems with unbounded controls María Soledad Aronna IMPA, Rio de Janeiro, Brazil (Joint work with M. Motta & F. Rampazzo, Università di Padova, Italy). Conference on New Trends in Optimal Control June 2014, Tours, France. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(3) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Outline For a CONTROL SYSTEM of the form ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. α=1. (x, u)(0) = (x̄, ū), with x : [0, T ] → IR n , u : [0, T ] → U ⊂ IR m , v : [0, T ] → V ⊂ IR l , we rely on the notion of LIMIT SOLUTION, and we investigate whether minimum problems with L1 −controls are PROPER EXTENSIONS of regular problems with more regular controls (AC or BV). Motivation: optimality conditions, numerical methods, etc. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(4) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Outline For a CONTROL SYSTEM of the form ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. α=1. (x, u)(0) = (x̄, ū), with x : [0, T ] → IR n , u : [0, T ] → U ⊂ IR m , v : [0, T ] → V ⊂ IR l , we rely on the notion of LIMIT SOLUTION, and we investigate whether minimum problems with L1 −controls are PROPER EXTENSIONS of regular problems with more regular controls (AC or BV). Motivation: optimality conditions, numerical methods, etc. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(5) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Outline For a CONTROL SYSTEM of the form ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. α=1. (x, u)(0) = (x̄, ū), with x : [0, T ] → IR n , u : [0, T ] → U ⊂ IR m , v : [0, T ] → V ⊂ IR l , we rely on the notion of LIMIT SOLUTION, and we investigate whether minimum problems with L1 −controls are PROPER EXTENSIONS of regular problems with more regular controls (AC or BV). Motivation: optimality conditions, numerical methods, etc. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(6) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Outline For a CONTROL SYSTEM of the form ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. α=1. (x, u)(0) = (x̄, ū), with x : [0, T ] → IR n , u : [0, T ] → U ⊂ IR m , v : [0, T ] → V ⊂ IR l , we rely on the notion of LIMIT SOLUTION, and we investigate whether minimum problems with L1 −controls are PROPER EXTENSIONS of regular problems with more regular controls (AC or BV). Motivation: optimality conditions, numerical methods, etc. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(7) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Limit solutions Consider the Cauchy problem ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. for t ∈ [0, T ],. α=1. (x, u)(0) = (x̄, ū). Here (x̄, ū) ∈ IR n × U, u ∈ L1 ([0, T ]; U) having u(0) = ū, and v ∈ L1 ([0, T ]; V ). We say that x : [0, T ] → IR n is a LIMIT SOLUTION if, for every τ ∈ [0, T ], there exists (ukτ ) ⊂ AC ([0, T ]; U) such that ukτ (0) = ū and the corresponding Carathéodory solutions (xkτ ) are uniformly bounded and satisfy |(xkτ , ukτ )(τ ) − (x, u)(τ )| + ∥(xkτ , ukτ ) − (x, u)∥1 → 0, . .. ... when k → ∞.. . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(8) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Limit solutions Consider the Cauchy problem ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. for t ∈ [0, T ],. α=1. (x, u)(0) = (x̄, ū). Here (x̄, ū) ∈ IR n × U, u ∈ L1 ([0, T ]; U) having u(0) = ū, and v ∈ L1 ([0, T ]; V ). We say that x : [0, T ] → IR n is a LIMIT SOLUTION if, for every τ ∈ [0, T ], there exists (ukτ ) ⊂ AC ([0, T ]; U) such that ukτ (0) = ū and the corresponding Carathéodory solutions (xkτ ) are uniformly bounded and satisfy |(xkτ , ukτ )(τ ) − (x, u)(τ )| + ∥(xkτ , ukτ ) − (x, u)∥1 → 0, . .. ... when k → ∞.. . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(9) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Limit solutions Consider the Cauchy problem ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. for t ∈ [0, T ],. α=1. (x, u)(0) = (x̄, ū). Here (x̄, ū) ∈ IR n × U, u ∈ L1 ([0, T ]; U) having u(0) = ū, and v ∈ L1 ([0, T ]; V ). We say that x : [0, T ] → IR n is a LIMIT SOLUTION if, for every τ ∈ [0, T ], there exists (ukτ ) ⊂ AC ([0, T ]; U) such that ukτ (0) = ū and the corresponding Carathéodory solutions (xkτ ) are uniformly bounded and satisfy |(xkτ , ukτ )(τ ) − (x, u)(τ )| + ∥(xkτ , ukτ ) − (x, u)∥1 → 0, . .. ... when k → ∞.. . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(10) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Example: AC −reachable set ̸= L1 −reachable set Fix (x̄, ū) ∈ IR n × U. Observe that the inclusion RAC ⊆ R can be strict: ẋ1 = u̇, ẋ2 = −1 + x12 , (x1 ,x2 )(0) = (1, 1), u(0) = 1, with U = [0, 1], t ∈ [0, 1]. It is trivial to verify that if ũ is absolutely continuous then the corresponding trajectory x̃ verifies x̃2 (1) > 0. In particular, (0, 0, 0) ̸∈ RAC . { On the other hand, setting û(t) :=. 1 0. if t = 0, we get that if t ∈ ]0, 1],. (0, 0, 0) ∈ R. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(11) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Example: AC −reachable set ̸= L1 −reachable set Fix (x̄, ū) ∈ IR n × U. Observe that the inclusion RAC ⊆ R can be strict: ẋ1 = u̇, ẋ2 = −1 + x12 , (x1 ,x2 )(0) = (1, 1), u(0) = 1, with U = [0, 1], t ∈ [0, 1]. It is trivial to verify that if ũ is absolutely continuous then the corresponding trajectory x̃ verifies x̃2 (1) > 0. In particular, (0, 0, 0) ̸∈ RAC . { On the other hand, setting û(t) :=. 1 0. if t = 0, we get that if t ∈ ]0, 1],. (0, 0, 0) ∈ R. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(12) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Example: AC −reachable set ̸= L1 −reachable set Fix (x̄, ū) ∈ IR n × U. Observe that the inclusion RAC ⊆ R can be strict: ẋ1 = u̇, ẋ2 = −1 + x12 , (x1 ,x2 )(0) = (1, 1), u(0) = 1, with U = [0, 1], t ∈ [0, 1]. It is trivial to verify that if ũ is absolutely continuous then the corresponding trajectory x̃ verifies x̃2 (1) > 0. In particular, (0, 0, 0) ̸∈ RAC . { On the other hand, setting û(t) :=. 1 0. if t = 0, we get that if t ∈ ]0, 1],. (0, 0, 0) ∈ R. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(13) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Example: AC −reachable set ̸= L1 −reachable set Fix (x̄, ū) ∈ IR n × U. Observe that the inclusion RAC ⊆ R can be strict: ẋ1 = u̇, ẋ2 = −1 + x12 , (x1 ,x2 )(0) = (1, 1), u(0) = 1, with U = [0, 1], t ∈ [0, 1]. It is trivial to verify that if ũ is absolutely continuous then the corresponding trajectory x̃ verifies x̃2 (1) > 0. In particular, (0, 0, 0) ̸∈ RAC . { On the other hand, setting û(t) :=. 1 0. if t = 0, we get that if t ∈ ]0, 1],. (0, 0, 0) ∈ R. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(14) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension of minimum problems . Definition . Let E be a set and let F : E → IR be a function. A proper extension of a minimum problem inf F(e), e∈E. is a new minimum problem inf F̂(ê). ê∈Ê. on a set Ê endowed with a limit notion and such that there exists an injective map i : E → Ê verifying the following properties: (i) F̂(i(e)) = F(e) for all e ∈ E and, moreover, for every ê ∈ Ê there exists a sequence (ek ) in E such that, setting êk := i(ek ), one has ( ) lim êk , F̂(êk ) = (ê, F̂(ê)), k→∞. (ii) inf F(e) = inf F̂(ê). e∈E ê∈Ê .. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(15) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension of minimum problems . Definition . Let E be a set and let F : E → IR be a function. A proper extension of a minimum problem inf F(e), e∈E. is a new minimum problem inf F̂(ê). ê∈Ê. on a set Ê endowed with a limit notion and such that there exists an injective map i : E → Ê verifying the following properties: (i) F̂(i(e)) = F(e) for all e ∈ E and, moreover, for every ê ∈ Ê there exists a sequence (ek ) in E such that, setting êk := i(ek ), one has ( ) lim êk , F̂(êk ) = (ê, F̂(ê)), k→∞. (ii) inf F(e) = inf F̂(ê). e∈E ê∈Ê .. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(16) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension of minimum problems . Definition . Let E be a set and let F : E → IR be a function. A proper extension of a minimum problem inf F(e), e∈E. is a new minimum problem inf F̂(ê). ê∈Ê. on a set Ê endowed with a limit notion and such that there exists an injective map i : E → Ê verifying the following properties: (i) F̂(i(e)) = F(e) for all e ∈ E and, moreover, for every ê ∈ Ê there exists a sequence (ek ) in E such that, setting êk := i(ek ), one has ( ) lim êk , F̂(êk ) = (ê, F̂(ê)), k→∞. (ii) inf F(e) = inf F̂(ê). e∈E ê∈Ê .. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(17) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension with NO final constraints For ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. (x, u)(0) = (x̄, ū),. α=1. consider a cost function ψ : IR n × U → IR . Define the following optimal control problems depending on (x̄, ū) : ( ) (P) inf ψ (x, u)(T ) : (u, v ) ∈ L1 × L1 , u(0) = ū, x ∈ Σ[x̄, u, v ], ( ) (PAC ) inf ψ (x, u)(T ) : (u, v ) ∈ AC × L1 , u(0) = ū, x = x[x̄, u, v ], . Theorem . (P) is a proper extension of (PAC ) : (i) for every x limit solution associated to (u, v ), there exists a sequence (uk ) ⊂ AC , uk (0) = 0, and xk := x[x̄, uk , v ] such that ∥(x, u) − (xk , uk )∥1 → 0, and ( ) ( ) inf 1 ψ (x, u)(T ) . (ii) inf1 1 ψ (x, u)(T ) = (u,v )∈L ×L. . Consequently, R = RAC .. (u,v )∈AC ×L . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(18) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension with NO final constraints For ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. (x, u)(0) = (x̄, ū),. α=1. consider a cost function ψ : IR n × U → IR . Define the following optimal control problems depending on (x̄, ū) : ( ) (P) inf ψ (x, u)(T ) : (u, v ) ∈ L1 × L1 , u(0) = ū, x ∈ Σ[x̄, u, v ], ( ) (PAC ) inf ψ (x, u)(T ) : (u, v ) ∈ AC × L1 , u(0) = ū, x = x[x̄, u, v ], . Theorem . (P) is a proper extension of (PAC ) : (i) for every x limit solution associated to (u, v ), there exists a sequence (uk ) ⊂ AC , uk (0) = 0, and xk := x[x̄, uk , v ] such that ∥(x, u) − (xk , uk )∥1 → 0, and ( ) ( ) inf 1 ψ (x, u)(T ) . (ii) inf1 1 ψ (x, u)(T ) = (u,v )∈L ×L. . Consequently, R = RAC .. (u,v )∈AC ×L . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(19) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension with NO final constraints For ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. (x, u)(0) = (x̄, ū),. α=1. consider a cost function ψ : IR n × U → IR . Define the following optimal control problems depending on (x̄, ū) : ( ) (P) inf ψ (x, u)(T ) : (u, v ) ∈ L1 × L1 , u(0) = ū, x ∈ Σ[x̄, u, v ], ( ) (PAC ) inf ψ (x, u)(T ) : (u, v ) ∈ AC × L1 , u(0) = ū, x = x[x̄, u, v ], . Theorem . (P) is a proper extension of (PAC ) : (i) for every x limit solution associated to (u, v ), there exists a sequence (uk ) ⊂ AC , uk (0) = 0, and xk := x[x̄, uk , v ] such that ∥(x, u) − (xk , uk )∥1 → 0, and ( ) ( ) inf 1 ψ (x, u)(T ) . (ii) inf1 1 ψ (x, u)(T ) = (u,v )∈L ×L. . Consequently, R = RAC .. (u,v )∈AC ×L . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(20) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension with NO final constraints For ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. (x, u)(0) = (x̄, ū),. α=1. consider a cost function ψ : IR n × U → IR . Define the following optimal control problems depending on (x̄, ū) : ( ) (P) inf ψ (x, u)(T ) : (u, v ) ∈ L1 × L1 , u(0) = ū, x ∈ Σ[x̄, u, v ], ( ) (PAC ) inf ψ (x, u)(T ) : (u, v ) ∈ AC × L1 , u(0) = ū, x = x[x̄, u, v ], . Theorem . (P) is a proper extension of (PAC ) : (i) for every x limit solution associated to (u, v ), there exists a sequence (uk ) ⊂ AC , uk (0) = 0, and xk := x[x̄, uk , v ] such that ∥(x, u) − (xk , uk )∥1 → 0, and ( ) ( ) inf 1 ψ (x, u)(T ) . (ii) inf1 1 ψ (x, u)(T ) = (u,v )∈L ×L. . Consequently, R = RAC .. (u,v )∈AC ×L . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(21) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Proper extension with NO final constraints For ẋ = f (x, u, v ) +. m ∑. gα (x)u̇α ,. on [0, T ],. (x, u)(0) = (x̄, ū),. α=1. consider a cost function ψ : IR n × U → IR . Define the following optimal control problems depending on (x̄, ū) : ( ) (P) inf ψ (x, u)(T ) : (u, v ) ∈ L1 × L1 , u(0) = ū, x ∈ Σ[x̄, u, v ], ( ) (PAC ) inf ψ (x, u)(T ) : (u, v ) ∈ AC × L1 , u(0) = ū, x = x[x̄, u, v ], . Theorem . (P) is a proper extension of (PAC ) : (i) for every x limit solution associated to (u, v ), there exists a sequence (uk ) ⊂ AC , uk (0) = 0, and xk := x[x̄, uk , v ] such that ∥(x, u) − (xk , uk )∥1 → 0, and ( ) ( ) inf 1 ψ (x, u)(T ) . (ii) inf1 1 ψ (x, u)(T ) = (u,v )∈L ×L. . Consequently, R = RAC .. (u,v )∈AC ×L . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(22) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. BV controls - Recall: Graph completions • For regular u ∈ AC one can reparametrize time t(s) = φ0 (s) with φ0 : [0, 1] → [0, T ], and set φ(s) := u ◦ φ0 (s). The SPACE-TIME SYSTEM: ′ y0 (s) = φ′0 (s), m ∑ y ′ (s) = f (y (s), φ(s), ψ(s))φ′0 (s) + gα (y (s))φα ′ (s), α=1 (y0 , y )(0) = (0, x̄) ,. s ∈ [0, 1].. • For BV controls u, let (φ0 , φ) be a graph completion of u : (φ0 , φ) : [0, 1] → [0, T ] × U Lipschitz continuous such that, ∀t ∈ [0, T ], there exists s ∈ [0, 1] verifying (t, u(t)) = (φ0 , φ)(s). . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(23) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. BV controls - Recall: Graph completions • For regular u ∈ AC one can reparametrize time t(s) = φ0 (s) with φ0 : [0, 1] → [0, T ], and set φ(s) := u ◦ φ0 (s). The SPACE-TIME SYSTEM: ′ y0 (s) = φ′0 (s), m ∑ y ′ (s) = f (y (s), φ(s), ψ(s))φ′0 (s) + gα (y (s))φα ′ (s), α=1 (y0 , y )(0) = (0, x̄) ,. s ∈ [0, 1].. • For BV controls u, let (φ0 , φ) be a graph completion of u : (φ0 , φ) : [0, 1] → [0, T ] × U Lipschitz continuous such that, ∀t ∈ [0, T ], there exists s ∈ [0, 1] verifying (t, u(t)) = (φ0 , φ)(s). . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(24) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Recall: Graph completion solutions. Given (φ0 , φ) : [0, 1] → [0, T ] × U, let y be the solution of the SPACE-TIME system Graph completion solution: (possibly) set-valued map x : [0, T ] ⇒ IR n , t Z=⇒ x(t) := y ◦ φ−1 0 (t). Single-valued graph completion solution: Let σ : [0, T ] → [0, 1] be a right-inverse of φ0 having u(t) = φ ◦ σ(t), x : [0, T ] → IR n ,. x(t) := y ◦ σ(t),. for all t ∈ [0, T ].. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(25) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Recall: Graph completion solutions. Given (φ0 , φ) : [0, 1] → [0, T ] × U, let y be the solution of the SPACE-TIME system Graph completion solution: (possibly) set-valued map x : [0, T ] ⇒ IR n , t Z=⇒ x(t) := y ◦ φ−1 0 (t). Single-valued graph completion solution: Let σ : [0, T ] → [0, 1] be a right-inverse of φ0 having u(t) = φ ◦ σ(t), x : [0, T ] → IR n ,. x(t) := y ◦ σ(t),. for all t ∈ [0, T ].. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(26) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Optimal control problems with bounded variation controls ( ) K +T (Pgc [x̄, u, v ], ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK gc Recall: A simple limit solution x : [0, T ] → IR n is a BV-SIMPLE limit solution if (ukτ ) can be chosen independently of τ and the approximating inputs uk have equibounded variation. ( ) K (PBVS ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK BVS [x̄, u, v ], From [Aronna & Rampazzo, 2013] (Rampazzo’s presentation) we know that +T ΣK [x̄, u, v ] = ΣK gc BVS [x̄, u, v ] . Theorem . For every initial condition (x̄, ū) ∈ IR n × U, one has .. K K lim Val (Pgc )(x̄, ū) = lim Val (PBVS )(x̄, ū) = V (x̄, ū).. K →∞. K →∞. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(27) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Optimal control problems with bounded variation controls ( ) K +T (Pgc [x̄, u, v ], ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK gc Recall: A simple limit solution x : [0, T ] → IR n is a BV-SIMPLE limit solution if (ukτ ) can be chosen independently of τ and the approximating inputs uk have equibounded variation. ( ) K (PBVS ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK BVS [x̄, u, v ], From [Aronna & Rampazzo, 2013] (Rampazzo’s presentation) we know that +T ΣK [x̄, u, v ] = ΣK gc BVS [x̄, u, v ] . Theorem . For every initial condition (x̄, ū) ∈ IR n × U, one has .. K K lim Val (Pgc )(x̄, ū) = lim Val (PBVS )(x̄, ū) = V (x̄, ū).. K →∞. K →∞. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(28) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Optimal control problems with bounded variation controls ( ) K +T (Pgc [x̄, u, v ], ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK gc Recall: A simple limit solution x : [0, T ] → IR n is a BV-SIMPLE limit solution if (ukτ ) can be chosen independently of τ and the approximating inputs uk have equibounded variation. ( ) K (PBVS ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK BVS [x̄, u, v ], From [Aronna & Rampazzo, 2013] (Rampazzo’s presentation) we know that +T ΣK [x̄, u, v ] = ΣK gc BVS [x̄, u, v ] . Theorem . For every initial condition (x̄, ū) ∈ IR n × U, one has .. K K lim Val (Pgc )(x̄, ū) = lim Val (PBVS )(x̄, ū) = V (x̄, ū).. K →∞. K →∞. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(29) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Optimal control problems with bounded variation controls ( ) K +T (Pgc [x̄, u, v ], ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK gc Recall: A simple limit solution x : [0, T ] → IR n is a BV-SIMPLE limit solution if (ukτ ) can be chosen independently of τ and the approximating inputs uk have equibounded variation. ( ) K (PBVS ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK BVS [x̄, u, v ], From [Aronna & Rampazzo, 2013] (Rampazzo’s presentation) we know that +T ΣK [x̄, u, v ] = ΣK gc BVS [x̄, u, v ] . Theorem . For every initial condition (x̄, ū) ∈ IR n × U, one has .. K K lim Val (Pgc )(x̄, ū) = lim Val (PBVS )(x̄, ū) = V (x̄, ū).. K →∞. K →∞. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(30) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Optimal control problems with bounded variation controls ( ) K +T (Pgc [x̄, u, v ], ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK gc Recall: A simple limit solution x : [0, T ] → IR n is a BV-SIMPLE limit solution if (ukτ ) can be chosen independently of τ and the approximating inputs uk have equibounded variation. ( ) K (PBVS ) inf ψ (x, u)(T ) : (u, v ) ∈ BV K ×L1 , u(0) = ū, x ∈ ΣK BVS [x̄, u, v ], From [Aronna & Rampazzo, 2013] (Rampazzo’s presentation) we know that +T ΣK [x̄, u, v ] = ΣK gc BVS [x̄, u, v ] . Theorem . For every initial condition (x̄, ū) ∈ IR n × U, one has .. K K lim Val (Pgc )(x̄, ū) = lim Val (PBVS )(x̄, ū) = V (x̄, ū).. K →∞. K →∞. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(31) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. The problem with final constraints. Let S ⊂ IR n × U be a closed subset. Define the problems ( ) (P c ) inf ψ (x, u)(T ) : (u, v ) ∈ L1 × L1 , u(0) = ū, x ∈ Σ[x̄, u, v ], (x, u)(T ) ∈ S, ( ) c (PAC ) inf ψ (x, u)(T ) : (u, v ) ∈ AC × L1 , u(0) = ū, x = x[x̄, u, v ], (x, u)(T ) ∈ S.. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(32) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. The problem with final constraints. Let S ⊂ IR n × U be a closed subset. Define the problems ( ) (P c ) inf ψ (x, u)(T ) : (u, v ) ∈ L1 × L1 , u(0) = ū, x ∈ Σ[x̄, u, v ], (x, u)(T ) ∈ S, ( ) c (PAC ) inf ψ (x, u)(T ) : (u, v ) ∈ AC × L1 , u(0) = ū, x = x[x̄, u, v ], (x, u)(T ) ∈ S.. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(33) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U inf x2 (1), ẋ1 (t) = u̇(t), ẋ2 (t) = |x1 (t)|, (x1 , x2 )(0) = (1, 0),. (. u(0) = 1. ) (x1 , x2 , u)(1) ∈ S := {(0, 0)} ∪ (IR × [1, +∞[) × [0, 1], 1 4 u(t) ∈ U := [− , ] 3 3. For every input u ∈ AC , one has x2 (1) =. ∫1 0. |u(s)|ds > 0.. c ) ≥ 1. Hence, if (x1 , x2 , u)(1) ∈ S =⇒ x2 (1) ≥ 1, =⇒ Val(PAC. On the other hand, by implementing the impulsive control { 1, t = 0, c ). u(t) := =⇒ x2 (1) = 0 =⇒ Val(P c ) ≤ 0<1 ≤ Val(PAC 0, t ∈]0, 1], . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(34) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U inf x2 (1), ẋ1 (t) = u̇(t), ẋ2 (t) = |x1 (t)|, (x1 , x2 )(0) = (1, 0),. (. u(0) = 1. ) (x1 , x2 , u)(1) ∈ S := {(0, 0)} ∪ (IR × [1, +∞[) × [0, 1], 1 4 u(t) ∈ U := [− , ] 3 3. For every input u ∈ AC , one has x2 (1) =. ∫1 0. |u(s)|ds > 0.. c ) ≥ 1. Hence, if (x1 , x2 , u)(1) ∈ S =⇒ x2 (1) ≥ 1, =⇒ Val(PAC. On the other hand, by implementing the impulsive control { 1, t = 0, c ). u(t) := =⇒ x2 (1) = 0 =⇒ Val(P c ) ≤ 0<1 ≤ Val(PAC 0, t ∈]0, 1], . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(35) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U inf x2 (1), ẋ1 (t) = u̇(t), ẋ2 (t) = |x1 (t)|, (x1 , x2 )(0) = (1, 0),. (. u(0) = 1. ) (x1 , x2 , u)(1) ∈ S := {(0, 0)} ∪ (IR × [1, +∞[) × [0, 1], 1 4 u(t) ∈ U := [− , ] 3 3. For every input u ∈ AC , one has x2 (1) =. ∫1 0. |u(s)|ds > 0.. c ) ≥ 1. Hence, if (x1 , x2 , u)(1) ∈ S =⇒ x2 (1) ≥ 1, =⇒ Val(PAC. On the other hand, by implementing the impulsive control { 1, t = 0, c ). u(t) := =⇒ x2 (1) = 0 =⇒ Val(P c ) ≤ 0<1 ≤ Val(PAC 0, t ∈]0, 1], . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(36) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U inf x2 (1), ẋ1 (t) = u̇(t), ẋ2 (t) = |x1 (t)|, (x1 , x2 )(0) = (1, 0),. (. u(0) = 1. ) (x1 , x2 , u)(1) ∈ S := {(0, 0)} ∪ (IR × [1, +∞[) × [0, 1], 1 4 u(t) ∈ U := [− , ] 3 3. For every input u ∈ AC , one has x2 (1) =. ∫1 0. |u(s)|ds > 0.. c ) ≥ 1. Hence, if (x1 , x2 , u)(1) ∈ S =⇒ x2 (1) ≥ 1, =⇒ Val(PAC. On the other hand, by implementing the impulsive control { 1, t = 0, c ). u(t) := =⇒ x2 (1) = 0 =⇒ Val(P c ) ≤ 0<1 ≤ Val(PAC 0, t ∈]0, 1], . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(37) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U [M.S. Aronna, M. Motta & F. Rampazzo, 2014]. Theorem (A sufficient condition for proper extension in the presence of terminal constraints): Assume that S is a compact set contained in the interior of IR n × U, and that there exist some positive constants ρ, η such that: . (i) for each x ∈ Sρ = B(S, ρ)\S, and ∀(px , pu ) ∈ D ∗ dS (x, u), we have {⟨ ⟩ } m ∑ α min px , gα (x, u)w + ⟨pu , w ⟩ < −η; |w |≤1. α=1. Limiting gradient: Let Ω ⊂ IR k be an open set, F : Ω → IR be locally Lipschitz continuous. For x ∈ Ω define D ∗ F (y ) := {w ∈ IR k : w = lim ∇F (yk ), yk ∈ DIFF \{y }, lim yk = y }. (ii) + viability. c ). Then (P c ) is a proper extension of (PAC. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(38) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U [M.S. Aronna, M. Motta & F. Rampazzo, 2014]. Theorem (A sufficient condition for proper extension in the presence of terminal constraints): Assume that S is a compact set contained in the interior of IR n × U, and that there exist some positive constants ρ, η such that: . (i) for each x ∈ Sρ = B(S, ρ)\S, and ∀(px , pu ) ∈ D ∗ dS (x, u), we have {⟨ ⟩ } m ∑ α min px , gα (x, u)w + ⟨pu , w ⟩ < −η; |w |≤1. α=1. Limiting gradient: Let Ω ⊂ IR k be an open set, F : Ω → IR be locally Lipschitz continuous. For x ∈ Ω define D ∗ F (y ) := {w ∈ IR k : w = lim ∇F (yk ), yk ∈ DIFF \{y }, lim yk = y }. (ii) + viability. c ). Then (P c ) is a proper extension of (PAC. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(39) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U [M.S. Aronna, M. Motta & F. Rampazzo, 2014]. Theorem (A sufficient condition for proper extension in the presence of terminal constraints): Assume that S is a compact set contained in the interior of IR n × U, and that there exist some positive constants ρ, η such that: . (i) for each x ∈ Sρ = B(S, ρ)\S, and ∀(px , pu ) ∈ D ∗ dS (x, u), we have {⟨ ⟩ } m ∑ α min px , gα (x, u)w + ⟨pu , w ⟩ < −η; |w |≤1. α=1. Limiting gradient: Let Ω ⊂ IR k be an open set, F : Ω → IR be locally Lipschitz continuous. For x ∈ Ω define D ∗ F (y ) := {w ∈ IR k : w = lim ∇F (yk ), yk ∈ DIFF \{y }, lim yk = y }. (ii) + viability. c ). Then (P c ) is a proper extension of (PAC. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(40) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U [M.S. Aronna, M. Motta & F. Rampazzo, 2014]. Theorem (A sufficient condition for proper extension in the presence of terminal constraints): Assume that S is a compact set contained in the interior of IR n × U, and that there exist some positive constants ρ, η such that: . (i) for each x ∈ Sρ = B(S, ρ)\S, and ∀(px , pu ) ∈ D ∗ dS (x, u), we have {⟨ ⟩ } m ∑ α min px , gα (x, u)w + ⟨pu , w ⟩ < −η; |w |≤1. α=1. Limiting gradient: Let Ω ⊂ IR k be an open set, F : Ω → IR be locally Lipschitz continuous. For x ∈ Ω define D ∗ F (y ) := {w ∈ IR k : w = lim ∇F (yk ), yk ∈ DIFF \{y }, lim yk = y }. (ii) + viability. c ). Then (P c ) is a proper extension of (PAC. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(41) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Final constraints S ⊂ IR n × U [M.S. Aronna, M. Motta & F. Rampazzo, 2014]. Theorem (A sufficient condition for proper extension in the presence of terminal constraints): Assume that S is a compact set contained in the interior of IR n × U, and that there exist some positive constants ρ, η such that: . (i) for each x ∈ Sρ = B(S, ρ)\S, and ∀(px , pu ) ∈ D ∗ dS (x, u), we have {⟨ ⟩ } m ∑ α min px , gα (x, u)w + ⟨pu , w ⟩ < −η; |w |≤1. α=1. Limiting gradient: Let Ω ⊂ IR k be an open set, F : Ω → IR be locally Lipschitz continuous. For x ∈ Ω define D ∗ F (y ) := {w ∈ IR k : w = lim ∇F (yk ), yk ∈ DIFF \{y }, lim yk = y }. (ii) + viability. c ). Then (P c ) is a proper extension of (PAC. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(42) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Sketch of the proof. Let us consider (u, v ) ∈ L1 × L1 and an associated limit solution x feasible for the problem (P c ), i.e. having (x, u)(T ) ∈ S. By definition of limit solution, there exists a sequence (ûk ) ⊂ AC such that ûk (0) = ū and ∥(x̂k , ûk ) − (x, u)∥1 + |(x̂k , ûk )(T ) − (x, u)(T )| → 0 as k → +∞, . where x̂k = x[x̄, ûk , v ] are the corresponding Carathéodory solutions.. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(43) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Sketch of the proof. Let us consider (u, v ) ∈ L1 × L1 and an associated limit solution x feasible for the problem (P c ), i.e. having (x, u)(T ) ∈ S. By definition of limit solution, there exists a sequence (ûk ) ⊂ AC such that ûk (0) = ū and ∥(x̂k , ûk ) − (x, u)∥1 + |(x̂k , ûk )(T ) − (x, u)(T )| → 0 as k → +∞, . where x̂k = x[x̄, ûk , v ] are the corresponding Carathéodory solutions.. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(44) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Sketch of the proof In general, (x̂k , ûk )(T ) ∈ / S. Step 1. We construct (σk ) such that σk → 0,. |(x̂k , ûk )(T − σk ) − (x̂k , ûk )(T )| → 0.. Step 2. We modify (x̂k , ûk ) from t = T − σk in the following way: we apply this estimate for the minimum time problem which holds in view of the quick reachability condition: TS,η (y ) ≤. dS (y ) . c(η). See e.g. [Motta & Rampazzo, 2013] This way we get (x̃k , ũk ) : [T − σk , Tk ] → IR n × U, having (x̃k , ũk )(Tk ) ∈ S. Step 3. If Tk < T , then we use the viability to remain in S until time . . . . . . . . . . . . . . . t = T. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(45) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Sketch of the proof In general, (x̂k , ûk )(T ) ∈ / S. Step 1. We construct (σk ) such that σk → 0,. |(x̂k , ûk )(T − σk ) − (x̂k , ûk )(T )| → 0.. Step 2. We modify (x̂k , ûk ) from t = T − σk in the following way: we apply this estimate for the minimum time problem which holds in view of the quick reachability condition: TS,η (y ) ≤. dS (y ) . c(η). See e.g. [Motta & Rampazzo, 2013] This way we get (x̃k , ũk ) : [T − σk , Tk ] → IR n × U, having (x̃k , ũk )(Tk ) ∈ S. Step 3. If Tk < T , then we use the viability to remain in S until time . . . . . . . . . . . . . . . t = T. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(46) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Sketch of the proof In general, (x̂k , ûk )(T ) ∈ / S. Step 1. We construct (σk ) such that σk → 0,. |(x̂k , ûk )(T − σk ) − (x̂k , ûk )(T )| → 0.. Step 2. We modify (x̂k , ûk ) from t = T − σk in the following way: we apply this estimate for the minimum time problem which holds in view of the quick reachability condition: TS,η (y ) ≤. dS (y ) . c(η). See e.g. [Motta & Rampazzo, 2013] This way we get (x̃k , ũk ) : [T − σk , Tk ] → IR n × U, having (x̃k , ũk )(Tk ) ∈ S. Step 3. If Tk < T , then we use the viability to remain in S until time . . . . . . . . . . . . . . . t = T. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(47) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Sketch of the proof In general, (x̂k , ûk )(T ) ∈ / S. Step 1. We construct (σk ) such that σk → 0,. |(x̂k , ûk )(T − σk ) − (x̂k , ûk )(T )| → 0.. Step 2. We modify (x̂k , ûk ) from t = T − σk in the following way: we apply this estimate for the minimum time problem which holds in view of the quick reachability condition: TS,η (y ) ≤. dS (y ) . c(η). See e.g. [Motta & Rampazzo, 2013] This way we get (x̃k , ũk ) : [T − σk , Tk ] → IR n × U, having (x̃k , ũk )(Tk ) ∈ S. Step 3. If Tk < T , then we use the viability to remain in S until time . . . . . . . . . . . . . . . t = T. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(48) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Sketch of the proof Sρ S. ) +(x, u)(T +(x̂k , x̂k )(T ) +(x , u )(T ) k k + (x̂k , ûk )(T − σk ) +(x , u )(T ) k. k. k. ■ . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(49) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Concluding remarks. The limit solutions naturally provide a proper extension of standard optimal control problems with no final constraints. The limit solutions optimal control problem (with L1 controls and trajectories) with no final constraints is the limit of problems with controls of variation K with K → ∞. When constraints are considered, a quick reachability condition + viability guarantee that the impulsive problem (with limit solutions as trajectories) provides a proper extension.. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(50) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Concluding remarks. The limit solutions naturally provide a proper extension of standard optimal control problems with no final constraints. The limit solutions optimal control problem (with L1 controls and trajectories) with no final constraints is the limit of problems with controls of variation K with K → ∞. When constraints are considered, a quick reachability condition + viability guarantee that the impulsive problem (with limit solutions as trajectories) provides a proper extension.. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(51) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. Concluding remarks. The limit solutions naturally provide a proper extension of standard optimal control problems with no final constraints. The limit solutions optimal control problem (with L1 controls and trajectories) with no final constraints is the limit of problems with controls of variation K with K → ∞. When constraints are considered, a quick reachability condition + viability guarantee that the impulsive problem (with limit solutions as trajectories) provides a proper extension.. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(52) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. References. M.S. Aronna & F. Rampazzo. In Proceedings of the 52nd IEEE Conference on Decision and Control, 2013. M.S. Aronna & F. Rampazzo. L1 limit solutions for control systems. 2013. Accepted for publication in JDE. M.S. Aronna & F. Rampazzo. A note on systems with ordinary and impulsive controls. To appear in IMA J. Math. Control Inform, 2014. M.S. Aronna, M. Motta & F. Rampazzo. Quick reachability and proper extension of unbounded control systems. [In preparation]. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
(53) Introduction & motivation. No terminal constraints. Controls with BV. Final constraints. THANK YOU FOR YOUR ATTENTION. . .. ... . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... . ... .. . . . .. .. ...
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