1
On the Problem of Minimum Asymptotic Exit Rate for Stochastically Perturbed arXiv:1405.1333v1 [math.DS] 6 May 2014
Multi-Channel Dynamical Systems Getachew K. Befekadu, and Panos J. Antsaklis,
Abstract We consider the problem of minimizing the asymptotic exit rate (from a given bounded open domain) for the state-trajectories of a multi-channel dynamical system with an asymptotically small random perturbation. In particular, for a class of admissible (reliable) stabilizing state-feedbacks, we show that the existence of a minimum asymptotic exit rate for the state-trajectories from the given domain is related to an asymptotic behavior (i.e., a probabilistic characterization) of the principal eigenvalue of the infinitesimal generator that corresponds to the stochastically perturbed dynamical system. Finally, we remark briefly on the implication of our result for evaluating the performance of the stabilizing state-feedbacks for the unperturbed multi-channel dynamical system.
Index Terms Asymptotic exit rate, diffusion equation, multi-channel dynamical systems, reliable systems.
I. I NTRODUCTION In this brief paper, we consider the problem of minimizing the asymptotic exit rate (from a given bounded open domain D ⊂ Rd ) pertaining to the diffusion process xǫ (t) that is described by the following Itˆo This work was supported in part by the National Science Foundation under Grant No. CNS-1035655. G. K. Befekadu is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail:
[email protected] P. J. Antsaklis is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail:
[email protected]
2
stochastic differential equation dxǫ (t) = Axǫ (t) +
where •
Xm
i=1
Bi ui (t) +
√
ǫσ(xǫ (t))dW (t), xǫ (0) = x0 , , ∀t ≥ 0,
(1)
A ∈ Rd×d , Bi ∈ Rd×ri , ǫ is a small positive number (which represents the level of random
perturbation in the system), •
σ : Rd → Rd×d is Lipschitz with the least eigenvalue of σ(·)σ T (·) uniformly bounded away from
zero, i.e., κ−1 kyk2 ≤ y T σ(x)σ T (x)y ≤ κkyk2 ,
∀x, y ∈ Rd ,
for some κ > 0. •
W (·) is a d-dimensional standard Wiener process, and
•
ui (·) is a Ui -valued measurable control process (i.e., an admissible control from the measurable set Ui ⊂ Rri ×d ) such that for all t > s, W (t) − W (s) is independent of ui (ν) for ν ≤ s and for all i = 1, 2, . . . , m.
In what follows, we consider a class of admissible controls ui (·) ∈ Ui of the form ui (t) = Ki xǫ (t), ∀t ≥ 0, with Ki ∈ Rri ×d for i = 1, 2, . . . , m such that1 Xm Sp A + Bi Ki ⊂ C− i=1
(2)
and
X Sp A +
i6=j
Bi Ki ⊂ C− ,
j = 1, 2, . . . , m.
Here, we remark that such a class of stabilizing state-feedbacks ( ) Ym ri ×d K⊆ K1 , K2 , . . . , Km ∈ R Equations (2) and (3) hold , i=1 | {z }
(3)
(4)
,K
can be linked to the problem of (reliable) stabilization for the unperturbed multi-channel dynamical systems, when there is a single failure in any of the control input channels (e.g. see [1] for a detailed characterization of such a class of stabilizing state-feedbacks). Therefore, in this brief paper, we will assume that the class of admissible (reliable) stabilizing state-feedbacks K for the unperturbed multichannel system (i.e., when ǫ = 0) in Equation (1) is non-empty. 1
Sp(A) denotes the spectrum of a matrix A ∈ Rd×d , i.e., Sp(A) = s ∈ C | rank(A − sI) < d .
3
Note that, for sufficiently small ǫ > 0, the infinitesimal generator that corresponds to the stochastically perturbed dynamical system in Equation (1), with ui (t) = Ki xǫ (t), for i = 1, 2, . . . , m, is given by D E ǫ n o Xm LǫK (·)(x) = ▽(·), A + Bi Ki x + tr σ(x)σ T (x) ▽2 (·) , i=1 2
(5)
where such an infinitesimal generator is uniformly elliptic with bounded coefficients.
Let D ⊂ Rd be a bounded open domain with smooth boundary (i.e., ∂D is a manifold of class C 2 [0, ∞]) and, for sufficiently small ǫ > 0, let ǫ τD = inf t > 0 xǫ (t) ∈ ∂D
(6)
denote the first exit-time from the domain D , which also depends on the class of admissible (reliable) stabilizing state-feedbacks K and, in particular, on the behavior of the solutions of the unperturbed multi-channel dynamical system Xm dx0 (t) = A + Bi Ki x0 (t), x0 (0) = x0 . i=1
(7)
Moreover, denote by C0T ([0, T ], Rd ) the space of all continuous functions ϕ(t), 0 ≤ t ≤ T , with range
in Rd ; and, in this space, we define the following metric ρ0T (ϕ, ψ) = sup ϕ(t) − ψ(t) ,
(8)
0≤t≤T
when ϕ(t), ψ(t) belong to C0T ([0, T ], Rd ). If Φ is a subset of the space C0T ([0, T ], Rd ), then we
define d0T (ψ, Φ) = sup ρ0T (ψ(t), ϕ(t)).
(9)
ϕ∈Φ
In Section II, we provide estimates for the minimum asymptotic exit rate by considering the diffusion ǫ of the process xǫ (t), Pǫx0 corresponding to the infinitesimal generator −LǫK and the first exit time τD
state-trajectories from the domain D .2 In particular, we minimize the following quantity3 λǫK = − lim sup T →∞
ǫ 1 >T , log Pǫx0 τD T
(10)
where the probability Pǫx0 , for sufficiently small ǫ > 0, is conditioned on the initial point x0 ∈ D as well as on the class of admissible (reliable) stabilizing state-feedbacks K. If the domain D contains a finite 2
We remark that such a solution for Equation (1) is assumed to have continuous sample paths with probability one (see [2]
for additional information). 3
In general, an asymptotic estimate for the minimum exit rate is related with maximization of the mean exit of the state-
trajectories from the given domain D.
4
number of compact sets containing the ω -limiting sets of the unperturbed multi-channel system, then the asymptotic behavior of the principal eigenvalue tends to zero exponentially, i.e., log λǫK = o(ǫ−1 ) as ǫ → 0 (e.g., see [3], [4] or [5] for additional discussions). Moreover, the principal eigenvalue λǫK of the
infinitesimal generator −LǫK turns out to be the boundary value between those positive R < rK for which −1 ǫ ǫ ) < ∞ and those R > r ǫ Eǫx0 exp(ǫ−1 RτD K for which Ex0 exp(ǫ RτD ) = ∞ (cf. Proposition 1 as well as Remark 1), where rK is given by the following 1 K rK = lim sup inf S0T (ϕ(t)) ϕ(t) ∈ D ∪ ∂D, ∀t ∈ [0, T ] . d T →∞ ϕ(t)∈C0T ([0,T ],R ) T
(11)
In general, such an asymptotic exit rate involves minimizing the following action functional4
2 Z Xm
1 T dϕ(t) K
S0T (ϕ(t)) = − A+ Bi Ki ϕ(t)
dt,
i=1 2 0 dt
(12)
ϕ(0)=x0
where
2
T Xm Xm
dϕ(t) dϕ(t)
− A+ Bi Ki ϕ(t) = Bi Ki ϕ(t)
dt − A + i=1 i=1 dt −1 dϕ(t) Xm × σ(ϕ(t))σ T (ϕ(t)) − A+ Bi Ki ϕ(t) , i=1 dt
(13)
and ϕ(t) ∈ C0T ([0, T ], Rd ) is absolutely continuous.
Before concluding this section, it is worth mentioning that estimating the minimum asymptotic exit rate with which the state-trajectories xǫ (t) exit from the domain D is related to a singularly perturbed eigenvalue problem. For example, for sufficiently small ǫ > 0, the asymptotic behavior for the principal eigenvalue corresponding to the following eigenvalue problem −LǫK Φǫ(x0 ,K)
=
λǫK Φǫ(x0 ,K)
Φǫ(x0 ,K) = 0
on
∂D
in
D
(14)
has been studied in the past (e.g., see [5] or [6] in the context of an asymptotic behavior for the principal eigenfunction; and see [7] or [8] in the context of an asymptotic behavior for the equilibrium density). Specifically, the authors in [8] and [7] have provided some results about the regularity properties of the action functional in connection with the asymptotic behavior of the equilibrium density, when the latter (i.e., the asymptotic behavior of the equilibrium density) is linked with the exit problem from the domain of attraction with an exponentially stable critical point for the stochastically perturbed dynamical system (cf. [9] and [10]). 4
aka Large-deviation rate function.
5
II. M AIN R ESULTS In this section, we present our main results – where we relate the problem of minimizing the asymptotic exit rate of the state-trajectories xǫ (t) (from the domain D ) and the asymptotic behavior of the principal eigenvalue for the infinitesimal generator corresponding to the stochastically perturbed dynamical multichannel system. In what follows, we provide some well known results that will be useful in the sequel (see [11, pp. 329– 332] and [3]). K (ϕ(·)) is a lower semicontinuous, i.e., if ϕ (t) → ϕ(t) in the space Lemma 1: The action functional S0T n
C0T ([0, T ], Rd ). Then
K K (ϕ(t)) ≤ lim inf S0T (ϕn (t)). S0T n→∞
(15)
˜ be a compact set in Rd , and let α be a positive number. Denote by Φ ˜ the class Lemma 2: Let D D,α ˜ and S K (ϕ(t)) ≤ α. Then, Φ ˜ is a compact of all functions ϕ(t) ∈ C0T ([0, T ], Rd ) with ϕ(0) ∈ D 0T D,α
subset of C0T ([0, T ], Rd ).
K (ϕ(·)) over the set Φ consisting of all ϕ(t) ∈ C ([0, T ], Rd ) Corollary 1: Consider the functional S0T 0T K (ϕ(·)) attains a minimum on Φ. with ϕ(0) ∈ K , where K is compact set. Then, S0T
Next, we state the following theorems that will be useful for proving our main results (see [3, Theorem 1.1, Theorem 1.2 and Lemma 9.1] or [12]; and see [11, pp. 332–340] for additional discussions). Theorem 1: For any α > 0, δ > 0 and γ > 0, there exists an ǫ0 > 0 such that o n n o K (ϕ(t)) + γ , ∀ǫ ∈ (0, ǫ0 ), Pǫx0 ρ0T xǫ (t), ϕ(t) < δ ≥ exp −ǫ−1 S0T
(16)
Theorem 2: For any α > 0, δ > 0 and γ > 0, there exists an ǫ0 > 0 such that o n n o Pǫx0 d0T xǫ (t), Φx0 ,α ≥ δ ≤ exp −ǫ−1 α − γ , ∀ǫ ∈ (0, ǫ0 ),
(17)
K (ϕ(t)) < α and ϕ(0) = x . where ϕ(t) is any function in C0T ([0, T ], Rd ) for which S0T 0
where
n o K Φx0 ,α = ϕ(t) ∈ C0T ([0, T ], Rd ) ϕ(0) = x0 and S0T (ϕ(t)) < α .
(18)
6
Theorem 3: Let D+δ denote a δ-neighborhood of D and let D−δ denote the set of points in D at a distance greater than δ from the boundary ∂D . Then, for sufficiently small δ > 0, the following estimates K S0T (ϕ(t)) ϕ(t) ∈ D+δ ∪ ∂D+δ , ∀t ∈ [0, T ] , (19) inf d ϕ(t)∈C0T ([0,T ],R ) ϕ(0)=x0
and
∈ D−δ ∪ ∂D−δ , ∀t ∈ [0, T ] ,
(20)
can be made arbitrarily close to each other. Furthermore, the same holds for K inf S0T (ϕ(t)) ϕ(t) ∈ D±δ ∪ ∂D±δ , ∀t ∈ [0, T ] , d
(21)
inf
ϕ(t)∈C0T ([0,T ],Rd ) ϕ(0)=x0
K S0T (ϕ(t)) ϕ(t)
ϕ(t)∈C0T ([0,T ],R ) ϕ(0)=x, ϕ(T )=y
uniformly for any x, y ∈ D−δ .
The following proposition asserts the existence of the principal eigenvalue of the infinitesimal generator −LǫK .
Proposition 1: Let λǫK be the principal eigenvalue of the infinitesimal generator −LǫK with zero boundary condition on ∂D . Then, we have −ǫ log λǫK → rK
as
ǫ → 0,
(22)
where rK
1 K = lim sup inf S0T (ϕ(t)) ϕ(t) ∈ D ∪ ∂D, ∀t ∈ [0, T ] . d T →∞ ϕ(t)∈C0T ([0,T ],R ) T
(23)
ϕ(0)=x0
Proof: Suppose that rK exists.5 . Then, using Theorem 3, one can show that rK also satisfies the following rK = sup x,y∈D
(
) 1 K lim sup inf . S0T (ϕ(t)) ϕ(t) ∈ D ∪ ∂D, ∀t ∈ [0, T ] d T →∞ ϕ(t)∈C0T ([0,T ],R ) T
(24)
ϕ(0)=x, ϕ(T )=y
ǫ ) tends to infinity, when R > r . Next, let us show that, for sufficiently small ǫ > 0, Eǫx0 exp(ǫ−1 RτD K If we choose a positive κ which is smaller than (R − rK )/3 so that 1 K sup inf S0T (ϕ(t)) ϕ(t) ∈ D ∪ ∂D, ∀t ∈ [0, T ] < rK + κ , d x,y∈D ϕ(t)∈C0T ([0,T ],R ) T ϕ(0)=x, ϕ(T )=y
5
Note that the existence of such a limit for rK can be easily established (e.g., see [3]).
(25)
7
and, for sufficiently small δ > 0, K inf ϕ(t) ∈ D ∪ ∂D , ∀t ∈ [0, T ] < T r + 2 κ , S (ϕ(t)) K −δ −δ 0T d ϕ(t)∈C0T ([0,T ],R ) ϕ(0)=x, ϕ(T )=y
(26)
for all x, y ∈ D−δ . Then, if we further let α = T rK + 2κ and γ = T κ, from Theorem 1, there exits
an ǫ0 > 0 such that
K S0T (ϕ(t)) ≤ T rK + 2κ ,
ϕ(t) ∈ D−δ ∪ ∂D−δ , ∀t ∈ [0, T ],
(27)
for any x, y ∈ D−δ ; and, moreover, we have the following probability estimate o n n o ǫ ǫ ǫ ǫ x (t), ϕ(t) < δ , ρ > T, x (τ ) ∈ D ≥ P Pǫx τD 0T D−δ −δ x −δ
K ≥ exp −ǫ−1 S0T (ϕ(t)) + γ , ≥ exp −ǫ−1 T rK + 3κ , ∀ǫ ∈ (0, ǫ0 ),
where ϕ(T ) ∈ D−2δ .
Let us define the following random events n o ǫ ǫ A n = τD > nT, x (nT ) ∈ D −δ , −δ
(28)
(29)
for n ∈ N+ ∪ {0}. Then, from the strong Markov property, we have Pǫx An ≥ Eǫx χAn−1 Pǫx(n−1)T A1 , ≥ Pǫx An−1 inf Pǫy A1 , y∈D−δ
≥ exp −ǫ−1 nT rK + 3κ
Note that, for an arbitrary n, we have the following
∀ǫ ∈ (0, ǫ0 ).
ǫ ǫ Eǫx exp ǫ−1 RτD ≥ exp ǫ−1 RnT Pǫx τD > nT , −δ −δ ≥ exp −ǫ−1 nT R − rK − 3κ , ∀ǫ ∈ (0, ǫ0 ),
(30)
(31)
ǫ which tends to infinity as n → ∞, i.e., Eǫx exp ǫ−1 RτD =∞ −δ
ǫ On the other hand, let us show that if R < rK , then, for sufficiently small ǫ > 0, Eǫx exp ǫ−1 RτD < −δ
∞. For κ < (R − rK )/3, let us choose δ so that K inf S0T (ϕ(t)) ϕ(t) ∈ D+δ ∪ ∂D+δ , ∀t ∈ [0, T ] > T rK − 2κ . d ϕ(t)∈C0T ([0,T ],R ) ϕ(0)=x0
(32)
8
From Theorem 2, with α = T rK − 2κ and γT κ, there exists an ǫ0 > 0 such that the distance between the set of functions ψ(t), for 0 ≤ t ≤ T , entirely lying in D and any of the sets Φx0 ,α is at least a distance δ; and, hence, we have the following probability estimate o n n o ǫ > T ≤ Pǫx d0T xǫ (t), Φx0 ,α ≥ δ , Pǫx τD ≤ exp −ǫ−1 T rK − 3κ ,
for any x ∈ D .
Then, using the Markov property, we have the following o n ǫ > nT ≤ exp −ǫ−1 nT rK − 3κ , Pǫx τD
∀ǫ ∈ (0, ǫ0 ),
∀ǫ ∈ (0, ǫ0 ),
(33)
(34)
and
X∞ ǫ ǫ Eǫx exp ǫ−1 RτD ≤ exp ǫ−1 R(n + 1)T Pǫx nT < τD ≤ (n + 1)T , n=0 X∞ ǫ ≤ exp ǫ−1 R(n + 1)T Pǫx τD > nT , n=0 X∞ ≤ exp ǫ−1 RT exp −ǫ−1 nT R − rK + 3κ , ∀ǫ ∈ (0, ǫ0 ), n=0
(35)
ǫ which converges to a finite value, i.e., Eǫx exp ǫ−1 RτD < ∞. Hence, rK is a boundary for which ǫ ) is finite. Then, from Equation (33) (cf. Equation(28), we have Eǫx exp(ǫ−1 rK τD n o 1 ǫ > T ≤ ǫ−1 rK − 3κ , ∀ǫ ∈ (0, ǫ0 ), (36) − log Pǫx τD T
for any x ∈ D , where the left side corresponds to − log λǫK . This completes the proof.
Remark 1: The above proposition, i.e., Proposition 1, states that the asymptotic behavior of the principal eigenvalue λǫK tends to zero exponentially, i.e., log λǫK = o(ǫ−1 ) as ǫ → 0 (cf. Equation (34) and
Equation (28)). Furthermore, the principal eigenvalue λǫK of the infinitesimal generator −LǫK is exactly ǫ ) < ∞ and equal to the boundary value between those positive R < rK for which Eǫx exp(ǫ−1 RτD ǫ ) = ∞ for any x ∈ D .6 those R > rK for which Eǫx exp(ǫ−1 RτD Finally, we briefly comment on the implication of our main results – when one is also interested in
either evaluating the performance of the admissible (reliable) stabilizing state-feedbacks in the system 6
Note that λǫK implicitly depends on the admissible (reliable) stabilizing state-feedbacks.
9
(or finding a set of (sub)-optimal stabilizing state-feedbacks); while minimizing the asymptotic exit rate (from the domain D ) for the state-trajectories.7 We conclude this section with the following corollary that establishes a connection between the minimum asymptotic exit rate rK ∗ and a set of (sub)-optimal stabilizing state-feedbacks K ∗ ν ⊂ K for the
unperturbed multi-channel dynamical system.
Corollary 2: Suppose that Proposition 1 holds, then there exists at least one m-tuple state-feedbacks ∗ ) ∈ K that satisfies K ∗ , (K1∗ , K2∗ , . . . Km (
) 1 K inf K ∈ arg min lim . S0T (ϕ(t)) ϕ(t) ∈ D ∪ ∂D, ∀t ∈ [0, T ] T →∞ ϕ(t)∈C0T ([0,T ],Rd ) T K∈K ∗
(37)
ϕ(0)=x0
Remark 2: Here it is worth remarking that if the problem of reliable stabilization for the unperturbed multi-channel dynamical system in Equation (1) is solvable (with respect to the class of admissible (reliable) stabilizing state-feedbacks K). Then, Corollary 2 implies that there exists at least one stabilizing state-feedback matrix that gives a minimum asymptotic exit rate (from the domain D ) for the statetrajectories of the dynamical system.
ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under Grant No. CNS-1035655.
R EFERENCES [1] G. K. Befekadu, V. Gupta and P. J. Antsaklis, “A further remark on the problem of reliable stabilization using rectangular dilated LMIs,” IMA J. of Math. Contr. Inf., vol. 30, no. 4, 571–575, 2013. [2] I. I. Gikhman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York, 1972. [3] A. D. Ventcel and M. I. Freidlin, “On small random perturbations of dynamical systems,” Russian Math. Surv., vol. 25, no. 1, 1–55, 1970. 7
Note that Proposition 1 provides a result on the asymptotic exit rate with respect to the nominal operating condition. However,
following the same discussion, we can also estimate the minimum asymptotic exit rates from the domain D, when one of the control input channels is extracted from the system due to a failure.
10
[4] A. D. Ventcel, “On the asymptotic behavior of the largest eigenvalue of a second-order elliptic differential operator with smaller parameter in the higher derivatives,” Soviet Math. Dokl., vol. 13, 13–17, 1972. [5] M. V. Day, “On the exponential exit law in the small parameter exit problem,” Stochastics, vol. 8, 297–323, 1983. [6] A. Devinatz and A Friedman, “Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem,” J. Indiana Univ. Math., vol. 27, 143–157, 1978. [7] Y. Kifer, “On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles,” J. Diff. Equ., vol 37, 108–139, 1980. [8] M. V. Day, “Recent progress on the small parameter exit problem,” Stochastics, vol. 20, 121–150, 1987. [9] S. J. Sheu, “Some estimates of the transition density of a non-degenerate diffusion Markov process,” Ann. Probab., vol. 19, no. 2, 538–561, 1991. [10] M. V. Day and T. A. Darden, “Some regularity results on the Ventcel-Freidlin quasipotential function,” Appl. Math. Optim., vol. 13, 259–282, 1985. [11] A. Friedman, Stochastic differential equations and applications, Academic Press, vol. II, 1976. [12] A. D. Ventcel, “Limit theorems on large deviations for stochastic processes,” Theo. Prob. Appl., vol. 18, no. 4, 817–821, 1973.
