The document discusses the application of Lyapunov functions in the context of Markov processes and dynamic programming, focusing on stability, average cost, and solution methods for dynamic programming equations. It explores various examples such as the MM1 queue and the Ornstein-Uhlenbeck model to illustrate key concepts related to rate matrices, generators, and resolvent equations. The conclusions emphasize the existence of steady-state distributions and the conditions required for moments to be finite.

1. Lyapunov functions, value functions,
and performance bounds
Sean Meyn
Department of Electrical and Computer Engineering
University of Illinois
and the Coordinated Science Laboratory
Joint work with R. Tweedie, I. Kontoyiannis, and P. Mehta
Supported in part by NSF (ECS 05 23620, and prior funding), and AFOSR

2. Objectives
Nonlinear state space model ≡ (controlled) Markov process,
state process X
Typical form:
dX(t) = f (X(t), U (t)) dt + σ(X(t), U (t)) dW (t)
noise
control

3. Objectives
Nonlinear state space model ≡ (controlled) Markov process,
state process X
Typical form:
dX(t) = f (X(t), U (t)) dt + σ(X(t), U (t)) dW (t)
noise
control
Questions: For a given feedback law,
• Is the state process stable?
• Is the average cost finite? E[c(X(t), U (t))]
• Can we solve the DP equations? min c(x, u) + Du h∗ (x) = η ∗
u
• Can we approximate the average cost η∗? The value function h∗ ?

4. Outline
Markov Models P t (x, · ) − π f →0
sup Ex [SτC (f )] < ∞
C
π(f ) < ∞
Representations
Lyapunov Theory DV (x) ≤ −f (x) + bIC (x)
Conclusions

5. I
Markov Models

6. Notation
Markov chain: X = X(t) : t ≥ 0
Countable state space, X
Transition semigroup,
P t (x, y) = P X(s + t) = y X(s) = x , x, y ∈ X

7. Notation: Generators & Resolvents
Markov chain: X = X(t) : t ≥ 0
Countable state space, X
Transition semigroup,
P t (x, y) = P X(s + t) = y X(s) = x , x, y ∈ X
Generator: For some domain of functions h,
1
Dh (x) = lim E[h(X(s + t)) − h(X(s)) X(s) = x]
t→0 t
1 t
= lim (P h (x) − h(x))
t→0 t

8. Notation: Generators & Resolvents
Generator: For some domain of functions h,
1
Dh (x) = lim E[h(X(s + t)) − h(X(s)) X(s) = x]
t→0 t
1 t
= lim (P h (x) − h(x))
t→0 t
Rate matrix:
Dh (x) = Q(x, y)h(y) P t = eQt
y

9. α µ
Example: MM1 Queue

x + 1 Prob εα

Sample paths: X(t + ε) ≈ x − 1 Prob εµ


x Prob 1 − ε(α + µ)
Rate matrix:
 
−α α 0 0 0 0 ···
 µ −α − µ
 α 0 0 0 · · ·

 0
 µ −α − µ α 0 0 · · ·


Q= 0 0 µ −α − µ α 0 · · ·

 0
 0 0 µ −α − µ α · · ·

 0
 0 0 0 µ −α − µ · · ·

.
. .
. .
. .
. .
. .
.
. . . . . .

10. 2 2
σW = 0 σW = 1
Example: O-U Model
Sample paths: dX(t) = AX(t) dt + B dW (t)
A n × n, Bn × 1, W standard BM
2
Generator: Dh (x) = (Ax)T h (x) + B T h (x)B

11. 2 2
σW = 0 σW = 1
Example: O-U Model
Sample paths: dX(t) = AX(t) dt + B dW (t)
A n × n, Bn × 1, W standard BM
2
Generator: Dh (x) = (Ax)T h (x) + B T h (x)B
h quadratic, h(x) = 1 xT P x
2 h (x) = P x
2
h (x) = P
Dh (x) = 1 xT (P A + AT P )x + B T P B
2

12. Notation: Generators & Resolvents
Generator: For some domain of functions h,
1
Dh (x) = lim E[h(X(s + t)) − h(X(s)) X(s) = x]
t→0 t
1 t
= lim (P h (x) − h(x))
t→0 t
Rate matrix:
Dh (x) = Q(x, y)h(y) P t = eQt
y
Resolvent:
∞
Rα = e−αt P t
0

13. Notation: Generators & Resolvents
Generator: For some domain of functions h,
1
Dh (x) = lim E[h(X(s + t)) − h(X(s)) X(s) = x]
t→0 t
1 t
= lim (P h (x) − h(x))
t→0 t
Rate matrix:
Dh (x) = Q(x, y)h(y) P t = eQt
y
Resolvent: Resolvent equations:
∞
Rα = e−αt P t Rα = [ Iα − Q]−1
0
QRα = Rα Q = αRα − I

14. Notation: Generators & Resolvents
Motivation: Dynamic programming. For a cost function c,
hα (x) = Rα c (x) = Rα (x, y)c(y)
y∈X
∞
= eαt E[c(X(t)) X(0) = x] dt
0
Discounted-cost value function

15. Notation: Generators & Resolvents
Motivation: Dynamic programming. For a cost function c,
hα (x) = Rα c (x) = Rα (x, y)c(y)
y∈X
∞
= eαt E[c(X(t)) X(0) = x] dt
0
Discounted-cost value function
Resolvent equation = dynamic programming equation,
c + Dhα = αhα

16. Notation: Steady State Distribution
Invariant (probability) measure π: X is stationary. In particular,
X(t) ∼ π, t≥0

17. Notation: Steady State Distribution
Invariant (probability) measure π: X is stationary. In particular,
X(t) ∼ π, t≥0
Characterizations:
π(x)P t (x, y) = π(y)
x∈X
α π(x)Rα (x, y) = π(y), α > 0
x∈X
π(x)Q(x, y) = 0
x∈X y ∈X

18. Notation: Relative Value Function
Invariant measure π, cost function c , steady-state mean η
Relative value function:
∞
h(x) = E[c(X(t)) − η X(0) = x] dt
0

19. Notation: Relative Value Function
Invariant measure π, cost function c , steady-state mean η
Relative value function:
∞
h(x) = E[c(X(t)) − η X(0) = x] dt
0
Solution to Poisson’s equation (average-cost DP equation):
c + Dh = η

20. II
Representations
π ∝ ν[I − (R − s ⊗ ν)]−1
h = [I − (R − s ⊗ ν)]−1 c
˜

21. Irreducibility
ψ-Irreducibility:
ψ(y) > 0 =⇒ P X(t) reaches y X(0) = x > 0 all x
ψ(y) > 0 =⇒ R(x, y) > 0 all x

22. Small Functions and Small Measures
ψ-Irreducibility:
ψ(y) > 0 =⇒ P X(t) reaches y X(0) = x > 0 all x
ψ(y) > 0 =⇒ R(x, y) > 0 all x
Small functions and measures: For a function s and probability ν,
R(x, y) ≥ s(x)ν(y), x, y ∈ X ∞
R= e−t P t dt
0

23. Small Functions and Small Measures
ψ-Irreducibility:
ψ(y) > 0 =⇒ P X(t) reaches y X(0) = x > 0 all x
ψ(y) > 0 =⇒ R(x, y) > 0 all x
Small functions and measures: For a function s and probability ν,
R(x, y) ≥ s(x)ν(y), x, y ∈ X
Resolvent dominates rank-one matrix, R=
∞
e−t P t dt
0
R ≥s⊗ν

24. Small Functions and Small Measures
ψ-Irreducibility:
ψ(y) > 0 =⇒ P X(t) reaches y X(0) = x > 0 all x
ψ(y) > 0 =⇒ R(x, y) > 0 all x
Small functions and measures: For a function s and probability ν,
R(x, y) ≥ s(x)ν(y), x, y ∈ X
Resolvent dominates rank-one matrix, R=
∞
e−t P t dt
0
R ≥s⊗ν
ψ-Irreducibility justi es assumption: s(x) > 0 for all x
and WLOG, ν = δx ∗ , where ψ(x∗ ) > 0

25. α µ
Example: MM1 Queue
R(x, y) > 0 for all x and y (irreducible in usual sense)
Conclusion:
R(x, y ) ≥ s(x)ν(y )
where s(x) := R(x, 0)
ν := δ0

26. 2 2
σW = 0 σW = 1
Example: O-U Model
dX(t) = AX(t) dt + B dW (t)
R(0, . ) Gaussian
Full rank if and only if (A, B) is controllable.
Conclusion: Under controllability, for any m, there is ε s.t.,
R(x, A) ≥ s(x)ν(A) all x and A
where s(x) = ε I x ≤m
ν(A) uniform on x ≤m

27. Potential Matrix
∞
Potential matrix: G(x, y) = (R − s ⊗ ν)n (x, y)
n=0
G = [I − (R − s ⊗ ν)]−1

28. Representation of π
∞
Potential matrix: G(x, y) = (R − s ⊗ ν)n (x, y)
n=0
π ∝ νG
νG (y) = ν(x)G(x, y)
x∈X

29. Representation of h
∞
Potential matrix: G(x, y) = (R − s ⊗ ν)n (x, y)
n=0
h = RG c
˜ + constant
c(x) = c(x) − η
˜
G˜ (y) =
c G(x, y)˜(y)
c
η= π(x)c(x) y∈X
y∈X

30. Representation of h
∞
Potential matrix: G(x, y) = (R − s ⊗ ν)n (x, y)
n=0
h = RG c
˜ + constant
c(x) = c(x) − η
˜
G˜ (y) =
c G(x, y)˜(y)
c
η= π(x)c(x) y∈X
y∈X
If sum converges, then Poisson’s equation is solved:
c(x) + Dh (x) = η

31. III
Lyapunov Theory
P n (x, · ) − π f →0
sup Ex [SτC (f )] < ∞
C
π(f ) < ∞
∆V (x) ≤ −f (x) + bIC (x)

32. Lyapunov Functions
DV ≤ −g + bs

33. Lyapunov Functions
DV ≤ −g + bs
General assumptions: V : X → (0,∞)
g : X → [1, ∞)
b < ∞, s small
e.g., s (x) = IC (x), C nite

34. Lyapunov Bounds on G DV ≤ −g + bs
Resolvent equation gives RV − V ≤ −Rg + bRs

35. Lyapunov Bounds on G DV ≤ −g + bs
Resolvent equation gives RV − V ≤ −Rg + bRs
Since s⊗ν is non-negative,
−[I − (R − s ⊗ ν)]V ≤ RV − V ≤ −Rg + bRs
G−1

36. Lyapunov Bounds on G DV ≤ −g + bs
Resolvent equation gives RV − V ≤ −Rg + bRs
Since s⊗ν is non-negative,
−[I − (R − s ⊗ ν)]V ≤ RV − V ≤ −Rg + bRs
G−1
More positivity, V ≥ GRg − bGRs
Some algebra, GR = G(R − s ⊗ ν) + (Gs) ⊗ ν ≥ G − I
Gs ≤ 1

37. Lyapunov Bounds on G DV ≤ −g + bs
Resolvent equation gives RV − V ≤ −Rg + bRs
Since s⊗ν is non-negative,
−[I − (R − s ⊗ ν)]V ≤ RV − V ≤ −Rg + bRs
G−1
More positivity, V ≥ GRg − bGRs
Some algebra, GR = G(R − s ⊗ ν) + (Gs) ⊗ ν ≥ G − I
Gs ≤ 1
General bound: GRg ≤ V + 2b
Gg ≤ V + g + 2b

38. Existence of π DV ≤ −g + bs
Condition (V2) DV ≤ −1 + bs
Representation: π ∝ νG
Bound: GRg ≤ V + 2b =⇒ G(x, X) ≤ V (x) + 2b
Conclusion: π exists as a probability measure on X

39. Existence of moments DV ≤ −g + bs
Condition (V3) DV ≤ −g + bs
Representation: π ∝ νG
Bound: Gg ≤ V + g + 2b

40. Existence of moments DV ≤ −g + bs
Condition (V3) DV ≤ −g + bs
Representation: π ∝ νG
Bound: Gg ≤ V + g + 2b
Conclusion: π exists as a probability measure on X
and the steady-state mean is nite,
π(g) := π(x)g(x) ≤ b
x∈X

41. α α µ
Example: MM1 Queue ρ =
µ
Linear Lyapunov function, V (x) = x
∞
DV (x) = Q(x, y)y
y=0
= α(x + 1) + µ(x − 1) − (α + µ)x
= −(µ − α) x>0
Conclusion: (V2) holds if and only if ρ < 1

42. α α µ
Example: MM1 Queue ρ =
µ
QuadraticLyapunov function, V (x) = x 2
∞
DV (x) = Q(x, y)y 2
y=0
= α(x + 1)2 + µ(x − 1)2 − (α + µ)x2
= α(x2 + 2x + 1) + µ(x2 − 2x + 1)2 − (α + µ)x2
= −2(µ − α)x + α + µ
Conclusion: (V3) holds, g(x) = 1 + x
if and only if ρ < 1

43. 2 2
σW = 0 σW = 1
Example: O-U Model
dX(t) = AX(t) dt + B dW (t)
h quadratic, h(x) = 1 xT P x
2 h (x) = P x
2
h (x) = P
Dh (x) = 1 xT (P A + AT P )x + B T P B
2
Suppose that P > 0 solves the Lyapunov equation,
P A + AT P = -I

44. 2 2
σW = 0 σW = 1
Example: O-U Model
dX(t) = AX(t) dt + B dW (t)
h quadratic, h(x) = 1 xT P x
2 h (x) = P x
2
h (x) = P
Dh (x) = 1 xT (P A + AT P )x + B T P B
2
Suppose that P > 0 solves the Lyapunov equation,
P A + AT P = -I
Then (V3) follows from the identity,
2 2 2
Dh (x) = − 1 x
2 + σX , σX = B T P B

45. 2 2
σW = 0 σW = 1
Example: O-U Model
dX(t) = AX(t) dt + B dW (t)
The function h(x) = 1 xT P x solves Poisson’s equation,
2
1 2
Dh = −g + η g(x) = 2 x
2
η = σX
Suppose that P > 0 solves the Lyapunov equation,
P A + AT P = -I
Then (V3) follows from the identity,
2 2 2
Dh (x) = − 1 x
2 + σX , σX = B T P B

46. Poisson’s Equation DV ≤ −g + bs
Condition (V3) DV ≤ −g + bs
Representation: h = RG c
˜ + constant c(x) = c(x) − η
˜
Bound: RGg ≤ V + 2b =⇒ RGg (x) ≤ V (x) + 2b

47. Poisson’s Equation DV ≤ −g + bs
Condition (V3) DV ≤ −g + bs
Representation: h = RG c
˜ + constant c(x) = c(x) − η
˜
Bound: RGg ≤ V + 2b =⇒ RGg (x) ≤ V (x) + 2b
Conclusion: If c is bounded by g, then h is bounded,
h(x) ≤ V (x) + 2b

48. α α µ
Example: MM1 Queue ρ =
µ
Poisson’s equation with g (x) = x
Dh = −g + η
We have (V3) with V a quadratic function of x:
Recall, with h (x) = x 2
Dh (x) = −2(µ − α)x + α + µ x>0

49. α α µ
Example: MM1 Queue ρ =
µ
Poisson’s equation with g (x) = x
Dh = −g + η
Solved with
x2 + x ρ
h(x) = 1
2 µ−α η=
1−ρ

50. IV
Conclusions

51. P t (x, · ) − π f →0
sup Ex [SτC (f )] < ∞
C
Final words
π(f ) < ∞
DV (x) ≤ −f (x) + bIC (x)
Just as in linear systems theory, Lyapunov functions
provide a characterization of system properties, as well
as a practical verification tool

52. P t (x, · ) − π f →0
sup Ex [SτC (f )] < ∞
C
Final words
π(f ) < ∞
DV (x) ≤ −f (x) + bIC (x)
Just as in linear systems theory, Lyapunov functions
provide a characterization of system properties, as well
as a practical verification tool
Much is left out of this survey - in particular,
• Converse theory
• Limit theory
• Approximation techniques to construct Lyapunov functions
or approximations to value functions
• Application to controlled Markov processes, and
approximate dynamic programming

53. References
[1,4] ψ-Irreducible foundations
[2,11,12,13] Mean- eld models, ODE models, and Lyapunov functions
[1,4,5,9,10] Operator-theoretic methods. See also appendix of [2]
[3,6,7,10] Generators and continuous time models
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