Converse linear quadratic optimal controlIt is well known that for a linear time invariant system (1) x ˙ = A x + B u with (A,B) controllable, there exists a static state feedback u = − K x such that the cost function (2) J = ∫ 0 ∞ x T Q x + u T R u d t is minimized, assuming Q ≥ 0 (positive semi-definite) and R &gt; 0 (positive definite). The gain K is the solution of the algebraic Riccati equation: 0 = A T P + P A − P B R − 1 B T P + Q K = R − 1 B T P P = P T ≥ 0 known as linear quadratic regulator (LQR). However, I wonder whether the converse also holds?That is, given a stabilizing K s (such that A − B K s is Hurwitz), do there exist matrices Q ≥ 0 and R &gt; 0 such that u = − K s x minimizes (2) given (1)? Or put differently:Question: Is every stabilizing linear state feedback optimal in some sense?

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Converse linear quadratic optimal controlIt is well known that for a linear time invariant system                    (1)                                          x            ˙                          =        A        x        +        B        u            with (A,B) controllable, there exists a static state feedback   u  =  −  K  x such that the cost function                    (2)                    J        =                  ∫          0                      ∞                                    x          T                Q        x        +                  u          T                R        u                d        t            is minimized, assuming   Q  ≥  0 (positive semi-definite) and   R  &amp;gt;  0 (positive definite). The gain K is the solution of the algebraic Riccati equation:                    0                            =                  A          T                P        +        P        A        −        P        B                  R                      −            1                                    B          T                P        +        Q                            K                            =                  R                      −            1                                    B          T                P                            P                            =                  P          T                ≥        0            known as linear quadratic regulator (LQR). However, I wonder whether the converse also holds?That is, given a stabilizing       K    s   (such that   A  −  B      K    s   is Hurwitz), do there exist matrices   Q  ≥  0 and   R  &amp;gt;  0 such that   u  =  −      K    s    x minimizes (2) given (1)? Or put differently:Question: Is every stabilizing linear state feedback optimal in some sense?

brulotfao · Accepted Answer

Step 1Given a system            x      ˙        =  A  x  +  B  u  ,    x  (      t    0    )  =      x    0    z  =      [                                        Q                          1                              /                            2                                                0                                      0                                      R                          1                              /                            2                                            ]        [                            x                                      u                      ]    ,with (A,B) is stabilizable, (Q,A) is detectable and   R  &amp;gt;  0 (positive definite). The linear quadratic regulator problem is given by minimizing      ∫          0              ∞            z    T    z  d  t  .Step 2From Boyd et al 1994: The inverse problem of optimal control is the following. Given a matrix K, determine if there exists   Q  ≥  0 and   R  &amp;gt;  0, such that (Q,A) is detectable and   u  =  K  x is the optimal control for the corresponding LQR problem. Equivalently, we seek   R  &amp;gt;  0 and   Q  ≥  0 such that there exists P nonnegative and P1 positive-definite satisfying  (  A  +  B  K      )    T    P  +  P  (  A  +  B  K  )  +      K    T    R  K  +  Q  =  0  ,        B    T    P  +  R  K  =  0and       A    T        P    1    +      P    1    A  &amp;lt;  Q. This is an LMIP in P,       P    1  , R and Q. (The condition involving       P    1   is equivalent to (Q,A) being detectable.)

Converse linear quadratic optimal control. It is well known that for a linear time invariant system dotx=Ax+Bu(1)

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