Finite-Horizon Discrete-Time Dynamic Systems

• Finite-horizon discrete-time dynamic systems are defined as systems evolving in discrete time steps over a fixed interval with explicit initial and terminal boundary conditions.
• The framework employs extended symplectic pencils and generalized Riccati equations to derive closed-form, non-recursive parametric solutions for optimal state and control trajectories.
• This methodology enhances robustness in optimal control design by accommodating singular matrices and varying dynamics, extending classical LQ techniques to descriptor systems.

A finite-horizon discrete-time dynamic system refers to any system or optimal control problem in which the evolution occurs in discrete time steps over a fixed, finite interval (e.g., $t = 0, \ldots, T$). In control theory, economics, and systems engineering, such systems are fundamental in the paper of optimization, planning, and feedback strategies where the future horizon is known and limited. The finite-horizon formulation is distinguished by two characteristics: explicit consideration of boundary conditions at both the start and end points and potential changes in system dynamics, costs, or constraints over the interval. This article surveys the mathematical structure, analysis, and application of finite-horizon discrete-time dynamic systems, particularly in the context of optimal control, with emphasis on techniques rooted in algebraic, geometric, and spectral decomposition, as advanced in recent research.

1. Algebraic Structure and the Extended Symplectic Pencil

A discrete-time, finite-horizon optimal control problem with quadratic cost and linear dynamics can be systematically encoded using an extended symplectic matrix pencil. This pencil, denoted $N-zM$, arises by expressing the set of all first-order optimality conditions (state evolution, co-state, and stationarity) as an implicit difference equation:

$M p(t+1) = N p(t), \quad t=0,\ldots,T-1,$

where $p(t)$ concatenates the state, input, and Lagrange multiplier (co-state) variables. The matrices $M$ and $N$ are constructed as:

$$M = \begin{bmatrix} I & -A^\top & 0 \ 0 & 0 & -B^\top \end{bmatrix}, \quad N = \begin{bmatrix} A & Q & S^\top \ B & S & R \end{bmatrix}.$$

The concept of the symplectic pencil is central in characterizing both feasible trajectories and optimal policy structures. This approach generalizes the "Hamiltonian" operator familiar from continuous-time optimal control, providing a compact algebraic mechanism to analyze coupled linear-quadratic dynamics over bounded horizons.

A significant result is that, given any solution $X$ to a generalized discrete-time Riccati equation (see below), there exist invertible matrices $U_X$, $V_X$ such that:

$$U_X (N-zM) V_X = \begin{bmatrix} A_X - zI & 0\ 0 & I - zA_K \ \end{bmatrix}$$

where $A_X$ denotes the closed-loop state matrix and $A_K$ relates to the dual eigenstructure. This block-diagonalization allows explicit parametrization of trajectories.

2. Parametric Solution for Optimal State and Control Trajectories

The decomposition of the extended symplectic pencil enables a closed-form, non-recursive parametric expression for the entire family of optimal state and control trajectories. For a partitioned state variable $x = [x_1; x_2]$, the dynamics for $x_2$ are

$x_2(t + 1) = A_{X, 22} x_2(t) + B_{12} u_2(t),$

which leads to an explicit solution involving initial and terminal conditions:

$$x_2(t) = A_{X,22}^{t} x_2(0) + \mathcal{A}_{B_{12}}'(T, t) \lambda_2(T),$$

$$u_2(t) = R_X^0 B_{12}^\top (A_{X,22})^{T-t-1} \lambda_2(T),$$

with $\lambda_2(T)$ encoding the boundary multiplier ("costate"). For $x_1$, after controlling for effects from $x_2$, the remaining degrees of freedom are isolated via reachability subspaces and manifest in a (possible) nonuniqueness of the optimal control, parametrized as:

$$u_1 = R_1^+ \big(x_1(T) - A_{X,11}^T x_1(0) - R_2 E \big) + (I - R_1 R_1^+) v_1$$

where $R_1^+$ is the Moore-Penrose pseudo-inverse.

This parametrization formalism encapsulates all optimal solutions, including cases where uniqueness fails due to non-strict convexity or degenerate boundary conditions.

3. Generalized Riccati Equations and Mild Assumptions

A core analytic step is the solution of a generalized (constrained) discrete-time algebraic Riccati equation (CGDARE), formulated as:

$$X = A^\top X A - (A^\top X B + S) (R + B^\top X B)^{+} (B^\top X A + S^\top) + Q,$$

$$\text{with constraint} \quad \ker(R + B^\top X B) \subseteq \ker(A^\top X B + S),$$

where $(\cdot)^+$ is the Moore-Penrose pseudo-inverse. These equations relax the restrictive assumptions found in standard Riccati theory:

• The method does not require the regularity of the symplectic pencil, i.e., it holds even if $\det(N - zM)$ vanishes identically for some $z$.
• Modulus controllability (a type of stabilizability) is not required; solution existence suffices.

These relaxations are significant for applicability, especially in the presence of singular weight matrices $R$, state or input constraints, or boundary conditions that force singularities in the associated matrix pencils.

Ancillary results include characterization of reachability and output-nulling subspaces via solutions to CGDARE, as well as the invariance of the closed-loop map structure to the particular solution $X$.

4. Spectral Properties and Eigenstructure Interpretation

The eigenstructure of the extended symplectic pencil dictates the qualitative and quantitative behavior of finite-horizon solutions. In the regular (nonsingular) case, the determinant admits a factorization:

$$\det(N - zM) = \det(A_X - zI) \cdot \det(I - zA_K) \cdot \det(R_X),$$

implying:

• Generalized eigenvalues of the pencil correspond to eigenvalues of the closed-loop $A_X$,
• The spectrum exhibits reciprocal structure ($\det(I - zA_K)$ yields reciprocals of $A_X$'s non-zero eigenvalues),
• The multiplicity at infinity is governed by the algebraic structure of $R_X$.

In the singular case ($R_X$ singular), only the eigenvalues of $A_X$ on the quotient space $\mathbb{R}^n / R_X$ (i.e., those not "hidden" by restriction to the output-nulling subspace) appear as genuine eigenvalues of $N-zM$.

Examples in the literature demonstrate subtle phenomena such as closed-loop matrices with eigenvalues not manifesting in the pencil due to reachability degeneracies.

5. Relaxed Framework and Implications for LQ and Descriptor Systems

The analytic machinery based on the extended symplectic pencil and CGDARE supports a broad unification of finite-horizon problems, including:

• $LQ$ problems with singular weight matrices,
• Descriptor systems and general boundary constraints,
• Problems where standard dynamic programming does not apply due to singularities or lack of full controllability.

In these cases, the decomposition allows both theoretical tractability (via geometric and algebraic tools) and explicit algorithmic construction of controllers and admissible trajectories.

Furthermore, the nonrecursive, closed-form expression of all optimal trajectories (state and input) enables efficient computation and characterization, particularly essential for problems with two-sided (affine or linear) boundary constraints.

6. Broader Relevance in Finite-Horizon Discrete-Time Optimal Control

The principal methodological advance is the harmonization of classical Riccati approaches, symplectic geometry, and algebraic analysis in a single, comprehensive framework for discrete-time, finite-horizon problems. Key practical implications include:

• Broader applicability to engineered and physical systems that often feature singular or poorly conditioned cost matrices,
• Foundation for robust controller synthesis where optimality conditions must handle degenerate cases,
• Clear spectral-theoretic insight into how boundary conditions and system structure influence achievable closed-loop behavior.

The result is a methodology that significantly enlarges the class of systems and problem formulations amenable to explicit optimal control on finite discrete horizons, while retaining the ability to analyze stability, performance, and structural properties via algebraic and geometric constructs.