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We consider the classic problem of pole placement by state feedback. Our recent work [1] offered an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing gain matrix to deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. The method was adapted from the classic eigenstructure assignment algorithm of Kautsky, Nichols and van Dooren [2]. In this paper we employ this parametric formula to introduce an unconstrained nonlinear optimisation algorithm to obtain a gain matrix that delivers any desired pole placement with optimal robustness.

In this paper, we provide conditions for the solvability of LQ (linear-quadratic) optimal control problems in which the input function is not penalised in the performance index. In particular, we focus our attention on regular input functions, i.e., on the control functions that do not involvedistributions, which in general appear in the solution of singular problems.