Infinite-Horizon Sparse Optimal Control: Solution through a Finite-Horizon Subproblem and Its Receding-Horizon Implementation

1. Introduction

• Earlier sparse-control work mostly used finite horizons, while choosing a horizon beforehand is difficult and the optimality of a sparse self-triggered alternative is unclear.Repeatedly shifting a finite horizon can itself produce a non-sparse input.Source: §1 · p. 1

2. Considered Problem

• The paper considers discrete-time infinite-horizon sparse control, using a sum of input 1-norms to induce sparsity; α must exceed 1 when the original system has unit-circle eigenvalues.In that setting, the higher penalty on farther-future inputs is paired with a requirement of exponential stability for the closed-loop system.Source: §1 · p. 2, §2 · p. 3

1. Introduction

• A sufficiently long finite-horizon subproblem yields an infinite-horizon optimum because its optimal input becomes zero after a finite time and remains optimal when extended by zeros.The paper estimates the required horizon length and discusses an adaptive choice when the estimate is conservative.Source: §1 · p. 2
• The receding-horizon approach can produce a non-sparse input, while the proposed implementation guarantees an input that is optimal and sparse in the infinite horizon.The literature also includes a sparse self-triggered approach whose infinite-horizon optimality is unclear.Source: §1 · pp. 1–2

2. Considered Problem

• The problem is considered for feasible initial states, since bounded inputs can make it infeasible when the initial state lies outside the null controllable region.The paper restricts its analysis to initial states in that region for which a feasible solution has finite objective value.Source: §2 · p. 4

3. Finite-Horizon Subproblem

• The section introduces a finite-horizon subproblem whose optimal input can also be optimal for the infinite-horizon problem after zero extension.The paper develops the horizon-length condition needed for this property.Source: §3 · p. 4
• F(N) considers only the anti-stable subsystem, obtained by decomposing the plant into stable and anti-stable parts.The anti-stable part is defined by eigenvalues outside the unit circle; the stable part has eigenvalues inside it.Source: §3 · p. 4
• A feasible input can be extended to a longer finite horizon by appending zeros when the anti-stable state is zero at the original horizon endpoint.The anti-stable state then remains zero through the added horizon.Source: §3 · p. 5
• For any N>K, an optimal solution of F(N) is zero from k=K through N−1, and its zero extension is optimal for every longer finite horizon.K is the defined zero-input time.Source: §3 · p. 5

4. Solution of the Infinite-Horizon Problem

• Among finite-objective feasible inputs, there is another feasible input with finite support and objective value no greater than the original input's.This finite-support result is used to establish the existence of the zero-input time.Source: §4 · p. 6
• If the infinite-horizon problem has a feasible solution with finite objective value, then for N>K, an optimal solution of F(N) exists and its zero extension is optimal in P.K is the zero-input time defined from the problem's feasibility conditions.Source: §4 · p. 6
• A common zero-input time can be used throughout a state-space polyhedron when each vertex subproblem is feasible with objective value at most V.Along a feasible input's state trajectory, the zero-input time need not be increased.Source: §4 · p. 7

5. Adaptive Choice of the Horizon Length

• Because the computed K may be conservative, the horizon can be chosen adaptively so that the condition of Theorem 2 is satisfied.The text describes this as a more practical way to make zero extension meaningful.Source: §5 · p. 7
• An optimal solution meeting Theorem 2's costate condition remains optimal after zero extension in every longer finite horizon and in P.The supplied passage does not state the costate condition itself.Source: §5 · p. 7

6. Receding-Horizon Implementation

• The receding-horizon input is optimal in P and zero from k=K onward when the horizon N exceeds K.The result assumes P has a feasible solution with finite objective value.Source: §6 · p. 8
• A sparse finite-horizon sequence does not ensure a sparse receding-horizon input, because the applied first value at each step may be nonzero.The theorem's receding-horizon input can be viewed as a zero extension of the finite sequence through K−1.Source: §6 · p. 9
• With adaptively chosen horizons satisfying Theorem 2's condition, the receding-horizon input is optimal in P and zero from K onward.Here K is defined for the initial state.Source: §6 · p. 9
• Figure 2's naive receding-horizon input is not sparse, remaining nonzero through at least k=199.The sum of its input norms through k=199 is 41.8.Source: Figure 2 · p. 10, §6 · p. 10

Example 1

• With a fixed horizon N = 200, the receding-horizon input was not sparse, although the finite-horizon problem’s optimal input was zero for k = 5 through 91.The applied first value can be nonzero even when each finite-horizon sequence is sparse.Source: §7 · p. 11
• 35.7 versus 41.8: the proposed input is infinite-horizon optimal and becomes zero after k = 36, earlier than the guaranteed zero-input time K = 37.The 41.8 value is a lower bound for the objective achieved by the previous receding-horizon input.Source: §7 · p. 11, Figure 3 · p. 11

Example 2

• The spacecraft motion was modeled in the orbital plane, with radial and tangential axes and no perpendicular motion considered.The system was discretized with a 1-second sampling period; because its matrix had eigenvalues on the unit circle, α = 1.05 was chosen.Source: §7 · p. 12
• N = 172 reduced the spacecraft example’s computed zero-input-time horizon K = 400; both input components became zero after k = 32.The adaptively chosen horizon satisfied the required condition, and the resulting input was almost the same as the receding-horizon input.Source: Figure 4 · p. 12, §7 · p. 13

Conclusion

• Zero-extending an optimal finite-horizon solution with horizon N longer than the zero-input time K gives an optimal, sparse infinite-horizon input.The paper also discusses explicit computation of K and adaptive choice of N.Source: §8 · p. 13
• The proposed approach can be used with receding-horizon control while retaining an optimal and sparse infinite-horizon input.Source: §8 · p. 13
• The objective is the sum of input 1-norms; extending the approach to more general objective functions remains future work.Source: §8 · p. 13

A. Proof of the Sufficiency Part of Lemma 1

• The comparison considers any input and its associated state trajectory under the system dynamics.Source: A. Proof of the Sufficiency Part of… · p. 13
• The proof rewrites the objective comparison using the state equation and a change in summation order.Source: A. Proof of the Sufficiency Part of… · pp. 13–14
• The optimality conditions make the first term minimized at u∗, while the remaining terms are constant with respect to the state trajectory.Source: A. Proof of the Sufficiency Part of… · p. 14

B. Proof of Lemma 2

• For N > K, the proof compares the finite-horizon optimum with a feasible zero extension whose objective is bounded by V.Source: B. Proof of Lemma 2 · p. 14
• Among the listed intervals, at least one must have inputs small relative to α^k; otherwise the objective would exceed the bound V.Source: B. Proof of Lemma 2 · p. 14
• The resulting bounds on the costate-input terms support extending the optimal input by zeros and applying the optimality conditions to establish optimality.Source: B. Proof of Lemma 2 · p. 14

• Stabilizability of (A, B) implies controllability of the anti-stable pair (Aa, Ba), which has full row rank.Source: C. Proof of Lemma 3 · p. 15
• Finite objective value implies u(k) tends to zero, allowing a time Ku to be chosen after a run of small input values.The proof also uses convergence of the state and its anti-stable part under the feasible input.Source: C. Proof of Lemma 3 · p. 15
• The proof defines a modified input by adding perturbations δj over na steps and bounds the resulting input magnitude.The stated bound is ∥u(k)∥∞≤αk.Source: C. Proof of Lemma 3 · p. 15
• After Ku, the anti-stable state is zero and the stable state converges to zero, so the input is feasible.Source: C. Proof of Lemma 3 · p. 15
• Inequalities (9) and (10) establish that the modified input does not increase the objective value.The proof frames this as the remaining step in the objective comparison.Source: C. Proof of Lemma 3 · pp. 15–16