- Email: [email protected]
On an extended notion of controllability*
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
On an extended notion of controllability Monica Motta ∗ Franco Rampazzo ∗∗ ∗
Dipartimento di Matematica Pura e Applicata, Universit` a di Padova (ITALY) - e-mail: [email protected] ∗∗ Dipartimento di Matematica Pura e Applicata, Universit` a di Padova (ITALY) - e-mail: [email protected] Abstract: The notion of controllability has been already extended to the case when the trajectories approach the target asymptotically. Accordingly, we give a sufficient first order condition for the continuity of the Value Function V corresponding to an integral cost with a Lagrangian l ≥ 0. The fact that l is allowed to be zero outside the target makes the continuity of V possible even in situations where Petrov’s inwards pointing field condition is not verified. Keywords: Extended controllability, nonlinear control problems 1. INTRODUCTION Let us consider a nonlinear control system on Rn y(t) ˙ = f (y(t), α(t)),
y(0) = x
(1)
and a payoff Zt J(t, y, α) =
l(y(s), α(s)) ds.
(2)
0
The control α takes values in a compact subset A ⊂ Rm . In the sequel we will use A to denote the set of Borelmeasurable controls α : [0, +∞[→ A. We assume that the current cost l : Rn × A → R is continuous and verifies l(y, a) ≥ 0
∀(y, a) ∈ Rn × A.
(3)
We are also given a closed set C ⊂ Rn with compact boundary, called the target. We shall consider the problem of minimizing the functional J by means of trajectories that reach the target asymptotically. The fact that l is allowed be zero on some region is crucial: it means that the state can pass across this region without paying any cost. We point out that classical assumptions include that l is strictly positive. In fact, if this hypothesis is not verified, it may happen that other functions besides the Value Function V solve the corresponding HamiltonJacobi equation. Moreover, regularizations of the problem –obtained e.g. by taking ln > 0 converging to l or by adding a coercivity term– may fail to converge to the original problem (see e.g. Bardi and Capuzzo Dolcetta [1997], Motta [2004], Malisoff [2004] and the references therein). The vector field f is assumed to be just continuous, so that we allow lack of uniqueness for local solutions of (1) (for a fixed control α). ? This research was partially supported by the M.U.R.S.T. project ”Viscosity, metric, and control theoretic methods for nonlinear partial differential equations.” 0 49J15
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Let us remark that we do not assume any hypothesis on the relationship between the zeros’ set of l and C. In particular, we do not require l to be strictly positive outside C. . Let us set Cc = Rn \C. Let x ∈ Cc , α ∈ A and let y(·) be a solution to (1) corresponding to α (and defined on some interval [0, b[⊆ [0, +∞[). We let . t(y) = inf{t ≥ 0 : y(t) ∈ C} (4) . and call t(y) the exit–time from Cc . We set t(y) = +∞ if the target is not reached. For any x ∈ Cc and any control map α ∈ A, let Y (x, α) denote the set of solutions of (1) corresponding to some restriction of α to a sub-interval [0, b[⊆ [0, +∞[. Let us introduce the subset of admissible trajectories S(x, α) ⊂ Y (x, α) by setting . S(x, α) = {y(·) ∈ Y (x, α) : lim inf d(y(t)) = 0}, t→t− (y)
where for any z ∈ Rn , d(z) denotes the distance of z from the target C. For any x ∈ Cc let us consider the set of admissible controls, . A(x) = {α : S(x, α) 6= ∅} and let us define the Value Function . V(x) = inf inf J(t(y), y(·), α) (5) α∈A(x)
y(·)∈S(x,α)
(V(x) = +∞ if A(x) = ∅). We shall introduce a condition on the data, under which for any initial state x in a neighborhood of the target one can construct an admissible trajectory of system (1) that approaches C at least asymptotically, with a cost proportional to a suitable power of d(x). As a consequence, V turns out to be continuous around C (see Theorem 1). This property is crucial in the regularization and uniqueness questions mentioned above. In particular the continuity of V is used in Motta and Sartori [2011], where infinite horizon problems with a possibly vanishing Lagrangian are studied.
6811
10.3182/20110828-6-IT-1002.02801
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Hypothesis 1. (H0) The control vector field f and the Lagrangian l are continuous and such that for any R > 0 one has |f (x1 , a) − f (x2 , a)| ≤ ωf (|x1 − x2 |, R), |l(x1 , a) − l(x2 , a)| ≤ ωl (|x1 − x2 |, R)
Theorem 3. Assume hypotheses (H0), (H1). Then there exist σ ¯ > 0 and C > 0 such that V(x) ≤ Cd2−γ (x) ∀x ∈ Cσ¯ \ C. See Section 4 for the proof of this theorem.
∀|x1 |, |x2 | ≤ R, ∀a ∈ A, where ωf (·, R), ωl (·, R) are moduli of continuity. 2. MAIN RESULT For every x ∈ Cc , let π(x) the set of projections of x on C: . π(x) = {y ∈ C | |x − y| = d(y)} Moreover, for any r > 0, let . Cr = {x ∈ Rn : d(x) < r}. We shall consider the following extended controllability hypothesis. Hypothesis 2. (H1) There exist real numbers σ, ν > 0, γ ∈]0, 2[, and a continuous function g : Rn × A → [0, +∞[ such that, ∀(x, a) ∈ Cc × A, one has: • g(x, a) ≥ l(x, a), g(x, a) > 0; • ∀x ∈ Cσ \ C there exist a ∈ A and y ∈ π(x) verifying x − y 1−γ f (x, a) d (x) , ≤ −ν (6) |x − y| g(x, a) .
Remark 1. When C is the closure of an open set with C 1,1 , compact boundary, condition (6) is equivalent to f (x, a) ˜ lim sup ∇d(x) , <0 ∀¯ x ∈ ∂C, g(x, a) x→¯ x, x∈Cc . ˜ ˜ where d(x) = d2−γ (x). Indeed, in this case ∇d(x) = (2 − x−π(x) 1−γ γ) |x−π(x)| d (x). Incidentally, notice that hypothesis (6) can be rephrased as the supersolution condition ˜ ˜ H(x, ∇d(x)) ≥0 ∀x ∈ Cσ \ C, where we have set f (x, a) . ˜ H(x, p) = sup − p, − ν(2 − γ) . g(x, a) a∈A Remark 2. When l(x, a) > 0 for all (x, a) ∈ Cc × A, a natural choice for g in the assumptions above is given by g ≡ l. Remark 3. Condition (6) generalizes the classical Petrov’s controllability condition which states the existence of a vector field f (x, a) uniformly pointing inwards the target (see e.g. Sontag [1998], Cannarsa and Sinestrari [2004], Bressan and Piccoli [2007]). When the sole weaker hypothesis (6) is fulfilled, instead, the vector field f (x, a) may become tangent to ∂C as x approaches the target.
3. SPACE-DEPENDENT TIME RESCALING In order to prove Theorem 1, we need some preliminary tools concerning time rescaling by means of the function g.
Definition 1. Ω ⊂ Rn be any open, nonempty set, and let f˜(y, a) be a continuous control vector field defined on Ω × A → Rn . Let us consider the control dynamics y 0 = f˜(y, a) (7) Let g˜ : Ω × A → R be a positive, continuous function. The system f˜(z, a) z0 = (8) g˜(z, a) will be called the (˜ g -rate)-control equation (with respect to the control equation (7)). In particular, (7) is the (1–rate)-control equation. Lemma 4. Let t¯ > 0, x ∈ Ω and let α ∈ A. Let y : [0, t¯] → Ω be a solution of the (1–rate)-control equation (7) corresponding to the control α and such that y(0) = x . Let us set, for every t ∈ [0, t¯], Zt . . G(t) = g˜(y(τ ), α(τ )) dτ, s¯ = G(t¯) 0
and let us use G−1 : [0, s¯] → [0, t¯] to denote the inverse of G. Then the path . [0, s¯] 3 s 7→ z(s) = y ◦ G−1 (s) (⊂ Ω) is a solution of the (˜ g -rate) control equation (8) corre. sponding to the control β = α ◦ G−1 and verifying the initial condition z(0) = x. Conversely, let s¯ > 0 and let z : [0, s¯] → Ω be a solution of the (˜ g -rate) control equation (8) corresponding to a control β ∈ A and such that z(0) = x. Let us set, for every s ∈ [0, s¯], Zs 1 . . dσ, t¯ = K(¯ s) K(s) = g˜(z(σ), β(σ)) 0
and let K −1 : [0, t¯] → [0, s¯] denote the inverse of K. Then the path . [0, t¯] 3 y : t 7→ y(t) = z ◦ K −1 (t) (⊂ Ω) is a solution of the (1–rate)-control equation (7) corre. sponding to the control α = β ◦ K −1 and verifying the initial condition y(0) = x. Observe that A is made of Borel maps, so the superposition α ◦ G−1 belongs to A if and only if α ∈ A. The proof consists of trivial applications of the chain rule.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
In the sequel we denote the value function V with Vf,l in order to stress the functional dependence of V on the the pair (f, l).
Definition 2. Consider the functional Φ : (f˜, ˜l) 7→ Vf˜,˜l where f˜, ˜l are continuous functions of (x, a) ∈ Rn × A verifying the same hypotheses as f and l, respectively, and Vf˜,˜l is the corresponding asymptotic exit-time value function. Lemma 5. • Φ(·) is positive 0-homogeneous with respect to continuous functions. I.e., if k = k(x, a) > 0 is a continuous function, Vf˜,˜l = Vkf˜,k˜l . • Φ(·) is positive 1-homogeneous in the second variable, with respect to positive real numbers. I.e., if η ∈ ]0, +∞[, Vf˜,η˜l = ηVf˜,˜l . • For every dynamics f˜, ˜l 7→ Φ(f˜, ˜l) is monotone in the following sense: ˜l ≤ g˜ in Rn × A =⇒ V ˜ ˜ ≤ V ˜ . f ,˜ g f ,l
We set
. Tf˜ = Vf˜,1 . Notice that Tf˜ is the usual minimum time function for the dynamics f˜. Corollary 6. For any (f˜, ˜l) and any continuous function g˜ ≥ ˜l in Rn × A, Vf˜,˜l ≤ Vf˜,˜g ≤ T f˜ . g ˜
∀x1 , x2 ∈ Cσ \ Cµ , ∀a ∈ A, where Mf , Mg , ωf (·), ωg (·) denote the maxima and the moduli of continuity of f and g, respectively, in the compact set Cσ × A. Let us first suppose that these constants and moduli are global in Rn × A, so that any trajectory z˜x˜ (·) of the grate control system w.r.t. system (1) with initial point . x ˜ ∈ nRn \ Cµ2 , turns out to obe defined for any s ≤ s¯ = min 1, inf{s : z˜x˜ (s) ∈ C µ22 } since |˜ zx˜ (s) − x ˜| ≤ all s ≤ s¯.
for
2
Let x ∈ Cσ \ C, and let (µk )k∈N be a decreasing sequence in ]0, σ] such that limk→+∞ µk = 0. We shall construct a trajectory z : [0, s¯[→ Cσ \ C of the g-rate control system such that lim z(s) ∈ C.
s→¯ s
This will be achieved by constructing a sequence of instanttrajectory pairs (sk , zk (·)), such that: i) for each k, zk (·) is . defined on [sk−1 , sk ] (s0 = 0); and ii) one has lim sk = s¯,
k→+∞
zk ([sk−1 , sk ]) ⊂ Cµk \ Cµk+1 ,
zk (sk−1 ) ∈ ∂Cµk ,
zk (sk ) = zk+1 (sk ) ∈ ∂Cµk+1 .
The step k = 1. Let us construct z(·) by a recursive procedure. Notice that µ1 ≤ σ < 1 and it is not restrictive to assume x ∈ ∂Cµ1 , namely d(x) = µ1 . Set . . . ν x1 = x, s01 = s0 = 0, s11 = d(x1 )2−γ , A1 where A1 will be determined later, let ax1 be a control defined agreeing with hypothesis (H1) w.r.t. x1 , and choose a solution z11 : [s01 , s11 ] → Rn of the Cauchy problem f (z, ax1 ) dz = ds g(z, ax1 )
z(0) = x1 ,
which exists for all s ≥ 0 such that z11 (s) does not hit C µ22 , as observed below (9). . Set x11 = x1 , and, for any j > 1, proceed recursively by setting j
. X ν sj1 = d(xi1 ))2−γ . A 1 i=1
. xj1 = z1j−1 (s1j−1 )
4. PROOF In view of Corollary 2, in order to prove Theorem 1, it is sufficient to prove the following claim.
Mf ρ µ2
Denote with (axj , y1j ) an arbitrary pair in A × π(xj1 ) 1
verifying hypothesis (H1) w.r.t. xj1 and choose a solution j n z1j : [sj−1 1 , s1 ] → R of the Cauchy problem
Claim 7. There exist σ ¯ > 0 and C > 0 such that T f (x) ≤ Cd2−γ (x) ∀x ∈ Cσ¯ \ C.
f (z, axj ) dz 1 = ds g(z, axj )
g
Proof. In view of hypothesis (H1), for any µ ∈]0, σ[ one has . ρµ = inf g(x, a) > 0.
1
Let us set o n . j j1 = inf j ≥ 2 : z1j [sj−1 1 , s1 ] ∩ Cµ2 6= ∅
(x,a)∈(Cσ \Cµ )×A
It is not restrictive to assume that σ < 1. Hence for every µ ∈]0, σ], we have: f (x1 , a) f (x2 , a) g(x1 , a) − g(x2 , a) ≤ 1 1 Mf ωg (|x1 − x2 |) + ωf (|x1 − x2 |), (9) 2 ρ ρ µ µ f (x1 , a) 1 g(x1 , a) ≤ ρµ Mf
j z(sj−1 1 ) = x1 .
Let us redefine sj11 , by letting, if j1 < ∞, n o sj11 = min s ∈]sj11 −1 , sj11 ] : z1j1 (s) ∈ Cµ2 , while we set sj11 = +∞ in the event j1 = ∞. (We shall see that, in fact, one always has j1 < ∞). For every j ≤ j1 and every s ∈]s1j−1 , sj1 [, one has
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
dz j 1 d j |z1 (s) − y1j |2 = h 1 (s), z1j (s) − y1j i = 2 ds ds * + f z1j (s), axj 1 , z1j (s) − xj1 g z1j (s), axj 1 * j + f z1 (s), axj f xj1 , axj j j 1 1 − , x1 − y1 + g z1j (s), axj g xj1 , axj 1 1 + * j f x1 , axj 1 , xj1 − y1j ≤ + j g x1 , axj 1 2 Mf j (s1 − sj−1 1 ) ρµ2 Mf Mf j + 2 ωg ) d(xj1 ) (s1 − sj−1 1 ρµ2 ρ µ2 1 Mf j j−1 + ωf (s1 − s1 ) d(xj1 ) ρµ2 ρµ 2 −νdγ (xj1 ) ≤ β1j (A1 ) − ν dγ (xj1 ), 2−γ j ν ≤ where we used sj1 − sj−1 ) and set d(x 1 1 A1 2 2−2γ Mf ν . β1j (A1 ) = d(xj1 ) A1 ρµ2 2−γ 1−γ 1 Mf ν j + 2 Mf ωg d(x1 ) d(xj1 ) ρµ2 ρµ2 A1 2−γ 1−γ Mf ν 1 ωf d(xj1 ) d(xj1 ) + ρµ2 ρµ2 A1
and
lim
sk − sk−1 ≤
2 2−γ (µk ) . ν(2 − γ)
(15)
If we specify the sequence (µk )k∈N by setting 2 . ∀k ∈ N µk = k −( 2−γ ) d(x) we obtain that, for all k ∈ N, k X π sk = (sh − sh−1 ) ≤ d2−γ (x). ν(2 − γ) h=1
uniformly w.r. to j. In particular, there exists a real number A¯1 such that 1 (10) A¯1 > ν 2 β1j (A¯1 ) < . 2 . Therefore, choosing A1 = A¯1 one obtains 1 d j ν |z (s) − y1j |2 ≤ − dγ (xj1 ), 2 ds 1 2 j−1 j j for all s ∈]s1 , s1 [. By integrating on [sj−1 1 , s1 ], one obtains 2 ν2 ≤ 1 − d2 (xj1 ). d(xj+1 ) 1 A1 q . ν2 Hence, setting k1 = 1 − A , we have 1 (11)
Arguing by induction, it is now not difficult to show that µ2 ≤ d(x1 )j ≤ µ1 < 1 for all j = 1, . . . , j1 . From the above inequalities it follows also that the whole trajectory j {z1j (s) : s ∈ [sj−1 1 , s1 ]} belongs to the compact set Cσ . . . Let us set s1 = sj11 x2 = xj11 , and let z1 : [s0 , s1 ] be the path defined by . j z1 (s) = z1j (s) ∀j = 1, . . . , j1 , s ∈ [sj−1 1 , s1 ] In particular, one has z1 (s0 ) = x1 , z1 (s1 ) = x2 , d(x2 ) = µ2
(13)
and
β1j (A1 ) = 0
j j d(xj+1 1 ) ≤ k1 d(x1 ) ≤ (k1 ) d(x1 ) ∀j.
sj1 ≤
The generic step. In the first step we have constructed a trajectory z1 : [s0 , s1 ] → Cµ1 verifying (12) and (13). By iterating the process, for every k ≥ 1 one can construct a trajectory zk : [sk−1 , sk ] → Cµk such that zk (sk ) ∈ ∂Cµk+1 (14)
Now, if µ2 ≤ d(xj1 ) ≤ µ1 < 1, for all j = 1, . . . , j1 , it is easy to show that A1 →+∞
j1 X
+∞ X 2−γ ν (k1 )j−1 d(x1 ) A1 j=1 j=1 ν 1 2−γ = (µ1 ) A1 1 − (k1 )2−γ It is now straightforward to check that 2 2−γ s1 ≤ (µ1 ) . ν(2 − γ)
s1 =
(12)
In particular, being (sk )k∈N increasing, it has a limit, say s¯: π s¯ = lim sk d2−γ (x) . ≤ k→∞ ν(2 − γ) Let us define the path z : [0, s¯[ by setting . z|[sk−1 ,sk ] (·) = zk (·) ∀k ∈ N. Since all the construction is done in the compact set Cσ , the restriction that the constants Mf , Mg and the moduli ωf (·), ωg (·) are global in Rn × A can be a posteriori removed. At this point, the proof of the Claim is concluded, since, T f ≤ s¯. g
REFERENCES M. Bardi and I. Capuzzo Docetta (1997). Optimal control and viscosity solutions J. of Adv. Differential Equations. Ed. Birkh¨auser, Boston. A. Bressan and B. Piccoli (2007). Introduction to the mathematical theory of control. AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO. P. Cannarsa and C. Sinestrari (2004). Semiconcave functions, Hamilton-Jacobi equations and optimal control . Birkh¨ auser Boston, Inc., Boston. M. Malisoff (2004). Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing Lagrangians, eikonal equations, and shape-from-shading, NoDEA Nonlinear Differential Equations Appl., 11, 95–122. M. Motta (2004). Viscosity solutions of HJB equations with unbounded data and characteristic points, Appl. Math. Optim., 49, 1–26. M. Motta and C. Sartori (2011). On some infinite horizon cheap control problems with unbounded data. 18th IFAC World Congress, Milan, Italy
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
E.D. Sontag (1998). Mathematical control theory. Deterministic finite-dimensional systems. Second edition. Texts in Applied Mathematics, 6. Springer-Verlag, New York
6815
On an extended notion of controllability Monica Motta ∗ Franco Rampazzo ∗∗ ∗
Dipartimento di Matematica Pura e Applicata, Universit` a di Padova (ITALY) - e-mail: [email protected] ∗∗ Dipartimento di Matematica Pura e Applicata, Universit` a di Padova (ITALY) - e-mail: [email protected] Abstract: The notion of controllability has been already extended to the case when the trajectories approach the target asymptotically. Accordingly, we give a sufficient first order condition for the continuity of the Value Function V corresponding to an integral cost with a Lagrangian l ≥ 0. The fact that l is allowed to be zero outside the target makes the continuity of V possible even in situations where Petrov’s inwards pointing field condition is not verified. Keywords: Extended controllability, nonlinear control problems 1. INTRODUCTION Let us consider a nonlinear control system on Rn y(t) ˙ = f (y(t), α(t)),
y(0) = x
(1)
and a payoff Zt J(t, y, α) =
l(y(s), α(s)) ds.
(2)
0
The control α takes values in a compact subset A ⊂ Rm . In the sequel we will use A to denote the set of Borelmeasurable controls α : [0, +∞[→ A. We assume that the current cost l : Rn × A → R is continuous and verifies l(y, a) ≥ 0
∀(y, a) ∈ Rn × A.
(3)
We are also given a closed set C ⊂ Rn with compact boundary, called the target. We shall consider the problem of minimizing the functional J by means of trajectories that reach the target asymptotically. The fact that l is allowed be zero on some region is crucial: it means that the state can pass across this region without paying any cost. We point out that classical assumptions include that l is strictly positive. In fact, if this hypothesis is not verified, it may happen that other functions besides the Value Function V solve the corresponding HamiltonJacobi equation. Moreover, regularizations of the problem –obtained e.g. by taking ln > 0 converging to l or by adding a coercivity term– may fail to converge to the original problem (see e.g. Bardi and Capuzzo Dolcetta [1997], Motta [2004], Malisoff [2004] and the references therein). The vector field f is assumed to be just continuous, so that we allow lack of uniqueness for local solutions of (1) (for a fixed control α). ? This research was partially supported by the M.U.R.S.T. project ”Viscosity, metric, and control theoretic methods for nonlinear partial differential equations.” 0 49J15
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Let us remark that we do not assume any hypothesis on the relationship between the zeros’ set of l and C. In particular, we do not require l to be strictly positive outside C. . Let us set Cc = Rn \C. Let x ∈ Cc , α ∈ A and let y(·) be a solution to (1) corresponding to α (and defined on some interval [0, b[⊆ [0, +∞[). We let . t(y) = inf{t ≥ 0 : y(t) ∈ C} (4) . and call t(y) the exit–time from Cc . We set t(y) = +∞ if the target is not reached. For any x ∈ Cc and any control map α ∈ A, let Y (x, α) denote the set of solutions of (1) corresponding to some restriction of α to a sub-interval [0, b[⊆ [0, +∞[. Let us introduce the subset of admissible trajectories S(x, α) ⊂ Y (x, α) by setting . S(x, α) = {y(·) ∈ Y (x, α) : lim inf d(y(t)) = 0}, t→t− (y)
where for any z ∈ Rn , d(z) denotes the distance of z from the target C. For any x ∈ Cc let us consider the set of admissible controls, . A(x) = {α : S(x, α) 6= ∅} and let us define the Value Function . V(x) = inf inf J(t(y), y(·), α) (5) α∈A(x)
y(·)∈S(x,α)
(V(x) = +∞ if A(x) = ∅). We shall introduce a condition on the data, under which for any initial state x in a neighborhood of the target one can construct an admissible trajectory of system (1) that approaches C at least asymptotically, with a cost proportional to a suitable power of d(x). As a consequence, V turns out to be continuous around C (see Theorem 1). This property is crucial in the regularization and uniqueness questions mentioned above. In particular the continuity of V is used in Motta and Sartori [2011], where infinite horizon problems with a possibly vanishing Lagrangian are studied.
6811
10.3182/20110828-6-IT-1002.02801
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Hypothesis 1. (H0) The control vector field f and the Lagrangian l are continuous and such that for any R > 0 one has |f (x1 , a) − f (x2 , a)| ≤ ωf (|x1 − x2 |, R), |l(x1 , a) − l(x2 , a)| ≤ ωl (|x1 − x2 |, R)
Theorem 3. Assume hypotheses (H0), (H1). Then there exist σ ¯ > 0 and C > 0 such that V(x) ≤ Cd2−γ (x) ∀x ∈ Cσ¯ \ C. See Section 4 for the proof of this theorem.
∀|x1 |, |x2 | ≤ R, ∀a ∈ A, where ωf (·, R), ωl (·, R) are moduli of continuity. 2. MAIN RESULT For every x ∈ Cc , let π(x) the set of projections of x on C: . π(x) = {y ∈ C | |x − y| = d(y)} Moreover, for any r > 0, let . Cr = {x ∈ Rn : d(x) < r}. We shall consider the following extended controllability hypothesis. Hypothesis 2. (H1) There exist real numbers σ, ν > 0, γ ∈]0, 2[, and a continuous function g : Rn × A → [0, +∞[ such that, ∀(x, a) ∈ Cc × A, one has: • g(x, a) ≥ l(x, a), g(x, a) > 0; • ∀x ∈ Cσ \ C there exist a ∈ A and y ∈ π(x) verifying x − y 1−γ f (x, a) d (x) , ≤ −ν (6) |x − y| g(x, a) .
Remark 1. When C is the closure of an open set with C 1,1 , compact boundary, condition (6) is equivalent to f (x, a) ˜ lim sup ∇d(x) , <0 ∀¯ x ∈ ∂C, g(x, a) x→¯ x, x∈Cc . ˜ ˜ where d(x) = d2−γ (x). Indeed, in this case ∇d(x) = (2 − x−π(x) 1−γ γ) |x−π(x)| d (x). Incidentally, notice that hypothesis (6) can be rephrased as the supersolution condition ˜ ˜ H(x, ∇d(x)) ≥0 ∀x ∈ Cσ \ C, where we have set f (x, a) . ˜ H(x, p) = sup − p, − ν(2 − γ) . g(x, a) a∈A Remark 2. When l(x, a) > 0 for all (x, a) ∈ Cc × A, a natural choice for g in the assumptions above is given by g ≡ l. Remark 3. Condition (6) generalizes the classical Petrov’s controllability condition which states the existence of a vector field f (x, a) uniformly pointing inwards the target (see e.g. Sontag [1998], Cannarsa and Sinestrari [2004], Bressan and Piccoli [2007]). When the sole weaker hypothesis (6) is fulfilled, instead, the vector field f (x, a) may become tangent to ∂C as x approaches the target.
3. SPACE-DEPENDENT TIME RESCALING In order to prove Theorem 1, we need some preliminary tools concerning time rescaling by means of the function g.
Definition 1. Ω ⊂ Rn be any open, nonempty set, and let f˜(y, a) be a continuous control vector field defined on Ω × A → Rn . Let us consider the control dynamics y 0 = f˜(y, a) (7) Let g˜ : Ω × A → R be a positive, continuous function. The system f˜(z, a) z0 = (8) g˜(z, a) will be called the (˜ g -rate)-control equation (with respect to the control equation (7)). In particular, (7) is the (1–rate)-control equation. Lemma 4. Let t¯ > 0, x ∈ Ω and let α ∈ A. Let y : [0, t¯] → Ω be a solution of the (1–rate)-control equation (7) corresponding to the control α and such that y(0) = x . Let us set, for every t ∈ [0, t¯], Zt . . G(t) = g˜(y(τ ), α(τ )) dτ, s¯ = G(t¯) 0
and let us use G−1 : [0, s¯] → [0, t¯] to denote the inverse of G. Then the path . [0, s¯] 3 s 7→ z(s) = y ◦ G−1 (s) (⊂ Ω) is a solution of the (˜ g -rate) control equation (8) corre. sponding to the control β = α ◦ G−1 and verifying the initial condition z(0) = x. Conversely, let s¯ > 0 and let z : [0, s¯] → Ω be a solution of the (˜ g -rate) control equation (8) corresponding to a control β ∈ A and such that z(0) = x. Let us set, for every s ∈ [0, s¯], Zs 1 . . dσ, t¯ = K(¯ s) K(s) = g˜(z(σ), β(σ)) 0
and let K −1 : [0, t¯] → [0, s¯] denote the inverse of K. Then the path . [0, t¯] 3 y : t 7→ y(t) = z ◦ K −1 (t) (⊂ Ω) is a solution of the (1–rate)-control equation (7) corre. sponding to the control α = β ◦ K −1 and verifying the initial condition y(0) = x. Observe that A is made of Borel maps, so the superposition α ◦ G−1 belongs to A if and only if α ∈ A. The proof consists of trivial applications of the chain rule.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
In the sequel we denote the value function V with Vf,l in order to stress the functional dependence of V on the the pair (f, l).
Definition 2. Consider the functional Φ : (f˜, ˜l) 7→ Vf˜,˜l where f˜, ˜l are continuous functions of (x, a) ∈ Rn × A verifying the same hypotheses as f and l, respectively, and Vf˜,˜l is the corresponding asymptotic exit-time value function. Lemma 5. • Φ(·) is positive 0-homogeneous with respect to continuous functions. I.e., if k = k(x, a) > 0 is a continuous function, Vf˜,˜l = Vkf˜,k˜l . • Φ(·) is positive 1-homogeneous in the second variable, with respect to positive real numbers. I.e., if η ∈ ]0, +∞[, Vf˜,η˜l = ηVf˜,˜l . • For every dynamics f˜, ˜l 7→ Φ(f˜, ˜l) is monotone in the following sense: ˜l ≤ g˜ in Rn × A =⇒ V ˜ ˜ ≤ V ˜ . f ,˜ g f ,l
We set
. Tf˜ = Vf˜,1 . Notice that Tf˜ is the usual minimum time function for the dynamics f˜. Corollary 6. For any (f˜, ˜l) and any continuous function g˜ ≥ ˜l in Rn × A, Vf˜,˜l ≤ Vf˜,˜g ≤ T f˜ . g ˜
∀x1 , x2 ∈ Cσ \ Cµ , ∀a ∈ A, where Mf , Mg , ωf (·), ωg (·) denote the maxima and the moduli of continuity of f and g, respectively, in the compact set Cσ × A. Let us first suppose that these constants and moduli are global in Rn × A, so that any trajectory z˜x˜ (·) of the grate control system w.r.t. system (1) with initial point . x ˜ ∈ nRn \ Cµ2 , turns out to obe defined for any s ≤ s¯ = min 1, inf{s : z˜x˜ (s) ∈ C µ22 } since |˜ zx˜ (s) − x ˜| ≤ all s ≤ s¯.
for
2
Let x ∈ Cσ \ C, and let (µk )k∈N be a decreasing sequence in ]0, σ] such that limk→+∞ µk = 0. We shall construct a trajectory z : [0, s¯[→ Cσ \ C of the g-rate control system such that lim z(s) ∈ C.
s→¯ s
This will be achieved by constructing a sequence of instanttrajectory pairs (sk , zk (·)), such that: i) for each k, zk (·) is . defined on [sk−1 , sk ] (s0 = 0); and ii) one has lim sk = s¯,
k→+∞
zk ([sk−1 , sk ]) ⊂ Cµk \ Cµk+1 ,
zk (sk−1 ) ∈ ∂Cµk ,
zk (sk ) = zk+1 (sk ) ∈ ∂Cµk+1 .
The step k = 1. Let us construct z(·) by a recursive procedure. Notice that µ1 ≤ σ < 1 and it is not restrictive to assume x ∈ ∂Cµ1 , namely d(x) = µ1 . Set . . . ν x1 = x, s01 = s0 = 0, s11 = d(x1 )2−γ , A1 where A1 will be determined later, let ax1 be a control defined agreeing with hypothesis (H1) w.r.t. x1 , and choose a solution z11 : [s01 , s11 ] → Rn of the Cauchy problem f (z, ax1 ) dz = ds g(z, ax1 )
z(0) = x1 ,
which exists for all s ≥ 0 such that z11 (s) does not hit C µ22 , as observed below (9). . Set x11 = x1 , and, for any j > 1, proceed recursively by setting j
. X ν sj1 = d(xi1 ))2−γ . A 1 i=1
. xj1 = z1j−1 (s1j−1 )
4. PROOF In view of Corollary 2, in order to prove Theorem 1, it is sufficient to prove the following claim.
Mf ρ µ2
Denote with (axj , y1j ) an arbitrary pair in A × π(xj1 ) 1
verifying hypothesis (H1) w.r.t. xj1 and choose a solution j n z1j : [sj−1 1 , s1 ] → R of the Cauchy problem
Claim 7. There exist σ ¯ > 0 and C > 0 such that T f (x) ≤ Cd2−γ (x) ∀x ∈ Cσ¯ \ C.
f (z, axj ) dz 1 = ds g(z, axj )
g
Proof. In view of hypothesis (H1), for any µ ∈]0, σ[ one has . ρµ = inf g(x, a) > 0.
1
Let us set o n . j j1 = inf j ≥ 2 : z1j [sj−1 1 , s1 ] ∩ Cµ2 6= ∅
(x,a)∈(Cσ \Cµ )×A
It is not restrictive to assume that σ < 1. Hence for every µ ∈]0, σ], we have: f (x1 , a) f (x2 , a) g(x1 , a) − g(x2 , a) ≤ 1 1 Mf ωg (|x1 − x2 |) + ωf (|x1 − x2 |), (9) 2 ρ ρ µ µ f (x1 , a) 1 g(x1 , a) ≤ ρµ Mf
j z(sj−1 1 ) = x1 .
Let us redefine sj11 , by letting, if j1 < ∞, n o sj11 = min s ∈]sj11 −1 , sj11 ] : z1j1 (s) ∈ Cµ2 , while we set sj11 = +∞ in the event j1 = ∞. (We shall see that, in fact, one always has j1 < ∞). For every j ≤ j1 and every s ∈]s1j−1 , sj1 [, one has
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
dz j 1 d j |z1 (s) − y1j |2 = h 1 (s), z1j (s) − y1j i = 2 ds ds * + f z1j (s), axj 1 , z1j (s) − xj1 g z1j (s), axj 1 * j + f z1 (s), axj f xj1 , axj j j 1 1 − , x1 − y1 + g z1j (s), axj g xj1 , axj 1 1 + * j f x1 , axj 1 , xj1 − y1j ≤ + j g x1 , axj 1 2 Mf j (s1 − sj−1 1 ) ρµ2 Mf Mf j + 2 ωg ) d(xj1 ) (s1 − sj−1 1 ρµ2 ρ µ2 1 Mf j j−1 + ωf (s1 − s1 ) d(xj1 ) ρµ2 ρµ 2 −νdγ (xj1 ) ≤ β1j (A1 ) − ν dγ (xj1 ), 2−γ j ν ≤ where we used sj1 − sj−1 ) and set d(x 1 1 A1 2 2−2γ Mf ν . β1j (A1 ) = d(xj1 ) A1 ρµ2 2−γ 1−γ 1 Mf ν j + 2 Mf ωg d(x1 ) d(xj1 ) ρµ2 ρµ2 A1 2−γ 1−γ Mf ν 1 ωf d(xj1 ) d(xj1 ) + ρµ2 ρµ2 A1
and
lim
sk − sk−1 ≤
2 2−γ (µk ) . ν(2 − γ)
(15)
If we specify the sequence (µk )k∈N by setting 2 . ∀k ∈ N µk = k −( 2−γ ) d(x) we obtain that, for all k ∈ N, k X π sk = (sh − sh−1 ) ≤ d2−γ (x). ν(2 − γ) h=1
uniformly w.r. to j. In particular, there exists a real number A¯1 such that 1 (10) A¯1 > ν 2 β1j (A¯1 ) < . 2 . Therefore, choosing A1 = A¯1 one obtains 1 d j ν |z (s) − y1j |2 ≤ − dγ (xj1 ), 2 ds 1 2 j−1 j j for all s ∈]s1 , s1 [. By integrating on [sj−1 1 , s1 ], one obtains 2 ν2 ≤ 1 − d2 (xj1 ). d(xj+1 ) 1 A1 q . ν2 Hence, setting k1 = 1 − A , we have 1 (11)
Arguing by induction, it is now not difficult to show that µ2 ≤ d(x1 )j ≤ µ1 < 1 for all j = 1, . . . , j1 . From the above inequalities it follows also that the whole trajectory j {z1j (s) : s ∈ [sj−1 1 , s1 ]} belongs to the compact set Cσ . . . Let us set s1 = sj11 x2 = xj11 , and let z1 : [s0 , s1 ] be the path defined by . j z1 (s) = z1j (s) ∀j = 1, . . . , j1 , s ∈ [sj−1 1 , s1 ] In particular, one has z1 (s0 ) = x1 , z1 (s1 ) = x2 , d(x2 ) = µ2
(13)
and
β1j (A1 ) = 0
j j d(xj+1 1 ) ≤ k1 d(x1 ) ≤ (k1 ) d(x1 ) ∀j.
sj1 ≤
The generic step. In the first step we have constructed a trajectory z1 : [s0 , s1 ] → Cµ1 verifying (12) and (13). By iterating the process, for every k ≥ 1 one can construct a trajectory zk : [sk−1 , sk ] → Cµk such that zk (sk ) ∈ ∂Cµk+1 (14)
Now, if µ2 ≤ d(xj1 ) ≤ µ1 < 1, for all j = 1, . . . , j1 , it is easy to show that A1 →+∞
j1 X
+∞ X 2−γ ν (k1 )j−1 d(x1 ) A1 j=1 j=1 ν 1 2−γ = (µ1 ) A1 1 − (k1 )2−γ It is now straightforward to check that 2 2−γ s1 ≤ (µ1 ) . ν(2 − γ)
s1 =
(12)
In particular, being (sk )k∈N increasing, it has a limit, say s¯: π s¯ = lim sk d2−γ (x) . ≤ k→∞ ν(2 − γ) Let us define the path z : [0, s¯[ by setting . z|[sk−1 ,sk ] (·) = zk (·) ∀k ∈ N. Since all the construction is done in the compact set Cσ , the restriction that the constants Mf , Mg and the moduli ωf (·), ωg (·) are global in Rn × A can be a posteriori removed. At this point, the proof of the Claim is concluded, since, T f ≤ s¯. g
REFERENCES M. Bardi and I. Capuzzo Docetta (1997). Optimal control and viscosity solutions J. of Adv. Differential Equations. Ed. Birkh¨auser, Boston. A. Bressan and B. Piccoli (2007). Introduction to the mathematical theory of control. AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO. P. Cannarsa and C. Sinestrari (2004). Semiconcave functions, Hamilton-Jacobi equations and optimal control . Birkh¨ auser Boston, Inc., Boston. M. Malisoff (2004). Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing Lagrangians, eikonal equations, and shape-from-shading, NoDEA Nonlinear Differential Equations Appl., 11, 95–122. M. Motta (2004). Viscosity solutions of HJB equations with unbounded data and characteristic points, Appl. Math. Optim., 49, 1–26. M. Motta and C. Sartori (2011). On some infinite horizon cheap control problems with unbounded data. 18th IFAC World Congress, Milan, Italy
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E.D. Sontag (1998). Mathematical control theory. Deterministic finite-dimensional systems. Second edition. Texts in Applied Mathematics, 6. Springer-Verlag, New York
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