Under suitable conditions, discounted optimal trajectories are asymptotically stable at an optimal equilibrium, see [14, 15, 33]. More precisely, if we write the optimally controlled system in feedback form
x(k+ 1) =f(x, µ(x)) (9.1)
with optimal feedback law4 µ:X→ U, then the closed loop system (9.1) has an optimal equilibrium with certain stability properties.
Theorem 9.2, below, provides conditions for such a result. For its formulation and the subsequent considerations we need the following practical asymptotic stability definition.
Here KL denotes the set of all continuous functions µ : [0,∞)2 → [0,∞) such that r 7→
µ(t, r) is in K for all t≥0 and t7→µ(t, r) is strictly decreasing to 0 for all r >0.
Definition 9.1: For two numbers ∆ > δ > 0, an equilibrium (xe, ue) is called (δ, ∆)-practically asymptotically stable, if there exists a functionη ∈ KLsuch that all closed-loop trajectoriesx(k) withkx(0)−xek ≤∆ satisfy the inequality
kx(k)−xek ≤max{η(kx(0)−xek, k), δ} (9.2) for all k∈N0.
The following is [14, Corollary 4.3], which will be used in the proof of Theorem 9.3 below.
Theorem 9.2: For β∈(0,1), consider a strictlyx-dissipative discounted optimal control problem at an equilibrium (xe, ue)∈Y. Assume that the optimal value function Ve of the modified problem (3.3) satisfies Ve(x)≤α2(kx−xek) and
Ve(x)≤C inf
u∈U
`˜β(x, u) (9.3)
for allx ∈X withϑ≤ kx−xek ≤Θ for 0≤ϑ <Θ, a function α2 ∈ K∞, and a constant C≥1 satisfying
C <1/(1−β). (9.4)
Then, whenever α(Θ) > α2(ϑ)/β holds for α from (3.1), the optimal closed-loop system is (δ,∆)-practically asymptotically stable with δ =α−1(α2(ϑ)/β) and ∆ =α−12 (α(Θ)). If (9.3) holds for all x ∈ X, then the equilibrium is asymptotically stable for the optimally controlled system.
Next we formulate the main result of this section, which states that undiscounted strict dissipativity implies semiglobal practical asymptotic stability for the discounted optimal closed loop system with β close to 1, provided the optimal equilibrium does not lie at the boundary of the constraint set. While at first glance the next theorem appears similar to Theorem 4.4 in [14], there is a decisive difference: whereas in [14] assumptions on the dis-counted optimal control problem are made, here we make assumptions on theundiscounted problem, but still derive stability properties for the discounted optimal solutions. In the theorem and its proof, β is again a varying parameter, hence we explicitly denote β as a function argument. As in the last section, to simplify notation, we writexeand ue instead of xe(1) andue(1), respectively.
Theorem 9.3: Consider an optimal control problem satisfying Assumption 8.1 with op-timal equilibrium (xe, ue) ∈ intY. Assume that there exists ˜α ∈ K∞ with Ve(x,1) ≤
˜
α(kx−xek) for all x ∈ X. Then for all ∆ > δ > 0 there is ¯β < 1 such that for each β ∈( ¯β,1) the optimal equilibrium xe(β) is (δ,∆)-practically asymptotically stable for the discounted optimal closed-loop system.
4In discrete time, the existence of an optimal feedback follows from the existence of open loop optimal control sequencesu∗by dynamic programming techniques, cf. [5].
Proof: Under the assumptions of this theorem, it follows that xe is a globally asymptoti-cally stable equilibrium of the undiscounted optimal closed-loop system. This follows, e.g., by applying [20, Theorem 4.8] to the problem with modified stage cost ˜`from (8.3), observ-ing that Ve(x,1)≥α(kx−xek) holds for α from the strict dissipativity assumption. This theorem implies that there exists µ∈ KL such that the undiscounted optimal trajectory x∗ satisfies
kx∗(k)−xek ≤µ(kx∗(0)−xek, k)≤µ(α−1(x∗(0)), k).
Moreover, the strict (x, u)-dissipativity and non-negativity of ˜` imply that k(x∗(k)−xe, u∗(k)−ue)k ≤α−1(˜`(x∗(k), u∗(k),1)≤α−1(Ve(x∗(0),1))
for allk∈N. Hence, for any Θ>0 all undiscounted optimal trajectories withVe(x∗(0),1)≤ Θ are uniformly bounded. By fixing an arbitrary Θ>0 and considering only those initial conditions satisfyingVe(x∗(0),1)≤Θ, for the following considerations, we may thus without loss of generality assume thatY is bounded.
Theorem 8.2 now implies the existence of ˆβ <1 such that the discounted problem is strictly (x, u)-dissipative for allβ∈( ˆβ,1). We claim that from this it follows (for allβ sufficiently close to 1) that the assumptions of Theorem 9.2 hold with α2 = (C1 + 1)˜α for C1 > 0 specified below. Since Y×[ ˆβ,1] is bounded and ˜` is continuous, for any Θ>0 we obtain a boundMΘ with ˜`(x∗(k), u∗(k), β)≤MΘ for all k∈N.
For the subsequent estimates we use the fact that the definitions of the rotated cost func-tions imply the existence of constants C1>0,C2>0 andβ1 <1 such that the inequality
`(x, u, β)˜ ≤C1`(x, u,˜ 1) +C2(1−β)2 (9.5) holds for all β ∈[β1,1] and all x, u∈Y.
In order to see that (9.5) holds, we use the following facts from the proof of Theorem 8.2, taking into account that the multipliers µi vanish because (xe, ue) ∈ intY: given ε > 0, there existsβ1<1 such that on the set [β1,1]
• the map β7→(xe(β), ue(β)) is Lipschitz continuous, implying that we can choose β1 <1 with k(xe(β)−xe, ue(β)−ue)k ≤ε/2 for allβ ∈[β1,1].
• ∇(x,u)`(x˜ e(β), ue(β), β) = 0.
• ∇2(x,u)`(x˜ e(β), ue(β), β) is positive definite, uniformly inβ.
• (x, u, β)7→ ∇2(x,u)`(x, u, β) is continuous, hence bounded on˜
N ={(x, u, β)∈Y×R| k(x, u)−(xe(β), ue(β))k< ε, β ∈[β1,1]}.
Due to continuity, the second derivatives of `, f, and λ are also bounded on N. Hence, by choosing ε > 0 small enough, Taylor’s theorem implies the existence of C > 0 with
`(x, u, β)˜ ≤Ck(x, u)−(xe(β), ue(β))k2for all (x, u, β)∈ N. Using the Lipschitz dependence of (xe(β), ue(β)) onβ and denoting the corresponding Lipschitz constant byL, this implies
`(x, u, β)˜ ≤Ck(x−xe(β), u−ue(β))k2 ≤CL(1−β)2.
Hence, (9.5) holds on N with C2 = CL and C1 ≥ 0 arbitrary. On (Y×[β1,1])\ N, the inequalities ˜`(x, u, β)≥σ(k(x−xe(β), u−ue(β))k) andk(x−xe(β), u−ue(β))k ≥ε/2 imply that there exists m >0 with ˜`(x, u,1)≥m for all (x, u)∈Y. Moreover, the boundedness of Yimplies the existence of M >0 with ˜`(x, u, β)≤M for all (x, u)∈Yand β ∈[β1,1].
This implies (9.5) withC1 =M/mandC2 ≥0 arbitrary on (Y×[β1,1])\ N and thus (9.5) on the whole set Y×[β1,1].
Using (9.5), the boundedness of ˜`and the fact that by nonnegativity of ˜`we haveβk`˜≤`˜ for all k≥0 we can now estimate forx=x∗(0)
Ve(x, β) ≤ J(x, ue ∗, β) =
∞
X
k=0
βk`(x˜ ∗(k), u∗(k), β)
≤
∞
X
k=0
βk
C1`(x˜ ∗(k), u∗(k),1) +C2(1−β)2
= C1
∞
X
k=0
βk`(x˜ ∗(k), u∗(k),1) +C2(1−β)
≤ C1Ve(x,1) +C2(1−β) ≤ C1α(˜ kx−xek) +C2(1−β).
Now let Θ> ϑ >0 be arbitrary and consider the set
S(β) :={x∈X|α(kx−xe(β)k)≤Θ andC1α(˜ kx−xek) +C2(1−β)≥ϑ}.
This set is compact, contains all x ∈X with ϑ≤Ve(x, β) ≤Θ and for every β2 < 1 with C2(1−β2)< ϑit does not contain a ball aroundxe for allβ ∈[β2,1].
Thus, there existsκ >0 independent ofβ ∈[β2,1] such that minx∈S(β)α(˜ kx−xek)≥κ >0.
Hence, choosing β3 ∈[β2,1) such that C2(1−β3)≤κ/2 and
(C1+ 1)|α(˜ kx−xek)−α(˜ kx−xe(β)k)| ≤κ/2 for all x∈S(β) and allβ ∈[β3,1] we obtain
Ve(x, β)≤C1α(˜ kx−xek) +C2(1−β)
≤(C1+ 1/2)˜α(kx−xek)≤(C1+ 1)˜α(kx−xe(β)k) for all x ∈ X with ϑ ≤ Ve(x, β) ≤ Θ and all β ∈ [β3,1]. This shows the first inequality needed in the Assumptions of Theorem 9.2.
From strict dissipativity we know that ˜`(x, u, β) ≥ α(kx −xe(β)k). This implies that
`(x, u, β)˜ ≥κ for all x∈S(β). Moreover, by continuity of all involved functions there is a bound B >0 such that the inequality (C1+ 1) ˜α(kx−xe(β)k)/α(kx−xe(β)k)≤B holds for allx∈S(β) and all β ∈[β3,1]. Hence, by choosing ¯β ∈[β3,1) such that 1−β <1/B holds, for allβ ∈[ ¯β,1] we obtain
(1−β)Ve(x, β) ≤ (1−β)(C1+ 1)˜α(kx−xe(β)k) ≤ (1−β)Bα(kx−xe(β)k)
< α(kx−xe(β)k) ≤ `(x, u, β),˜
which implies the second inequality from the Assumptions of Theorem 9.2 with C=B <
1/(1−β). Hence, Theorem 9.2 applies and yields the claim.
10 Conclusions
Prior work in the literature demonstrated a close connection between strict dissipativity, available storage, the turnpike property, and the near optimality of closed-loop solutions of model predictive control schemes. These classical notions of dissipativity and available storage are related to an optimal control problem with an undiscounted stage cost. In this paper, we modified these classical notions for application to optimal control problems with a discounted stage cost and showed that an important class of problems, namely affine linear systems with a strictly convex stage cost, satisfy these modified notions.
We subsequently demonstrated that discounted strict dissipativity is equivalent to a form of robust optimality (Theorem 5.4) and that discounted strict dissipativity implies a cer-tain continuity of trajectories near an optimal equilibrium (Theorem 7.1). These results are a prerequisite to demonstrating an equivalence between discounted strict dissipativity, turnpike properties, and near optimality of closed loop solutions of model predictive control schemes based on optimal control problems with discounted stage costs.
Under certain regularity conditions commonly used in the context of nonlinear program-ming, we demonstrated that strict dissipativity implies discounted strict dissipativity for discount factors close enough to one. Hence, statements in economic model predictive con-trol about steady-state optimality, turnpike properties, and closed-loop performance and convergence, are preserved under sufficiently mild discounting. We additionally showed that, under standard assumptions solely on the undiscounted problem, optimal controls computed from a discounted stage cost yield a (practically) asymptotically stable equilib-rium in closed-loop, again for sufficiently mild discounting. Importantly, our motivating applications in economics usually have a discount factor of 0.95 or higher [13, 24, 31, 40].
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