Duality theory for the time-varying 4-block problem
GONG Ting, LU Yu-feng
1. School of Mathematics, Dongbei University of Finance and Economics, Dalian 116025, China;
2. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Abstract This paper solves the block problems which play a fundamental role in the optimal
control theory under the operator-theoretic framework. Speciˉc duality theory is established for 4-block
problem by computing the appropriate annihilator and preannihilator of subspaces in such optimal
problem. Existence of optimal controllers is ensured and formulas for the performance index are
derived. It is also shown that the known results about duality theories for time-varying 1-block and
2-block problems are both special cases of the presented results in this paper. Moreover, an example
is given to prove that the optimum obtained by duality theory for a compact plant is time-varying allpass.
