Introduction:

Reachability is a key property of linear systems in control theory and plays a central role in system analysis. The minimum energy control problem is closely linked to reachability: in a reachable system, there typically exist multiple admissible controls capable of driving the system from a given initial state to a desired final state within a specified time interval. Singular (or descriptor) systems, which commonly appear in applications such as electrical circuits, constrained mechanical systems, and economic models, introduce additional challenges due to algebraic constraints and the potential non-uniqueness of solutions.

Method:

The proposed approach is based on transforming the minimum energy control problem of a singular continuous-time linear system with rectangular input matrices into its canonical form using the Weierstrass–Kronecker decomposition of matrix pencils. This decomposition allows the separation of the dynamic and algebraic components of the system. The minimum energy control problem is then formulated and solved. We derive explicit conditions for the existence of admissible controls and construct an optimal control law. The analytical results are established using techniques from linear algebra and system theory.

Result:

The derived conditions provide a systematic procedure for computing the optimal input and the corresponding minimum value of the performance index. The effectiveness of the proposed approach is illustrated through a numerical example, demonstrating its applicability to various classes of singular systems.

Conclusion:

This work presents a systematic method for the optimal control of singular systems based on the Weierstrass–Kronecker decomposition. The approach simplifies the analysis of constrained dynamics and provides explicit control strategies.