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In the paper, we first make a classification for general finite-horizon time-varying linear quadratic(FH-LQ) problems and give out a necessary and sufficient condition on which the problem belongs to some class (Them 2). We prove that for the FH-LQ problems the statements of finite infimum and achieved finite infrmum-are equivalent to each other (Them l). Basing on this we derive some new conditions for the regular case (Them 3). Finally, we discuss the FH-LQ problem with control energy constraint via an approach called "lossless S-procedure".
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