Linear–State Control Problems and Differential Games: Deterministic and Stochastic Systems

Abstract This paper concerns a class of linear-state optimal control problems and noncooperative differential games. Deterministic and stochastic systems are considered, as well as finite- and infinite-horizon problems. We give conditions under which these systems have degenerate feedback optimal controls so that the optimal control actions $$a(t,x) \equiv a(t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≡</mml:mo> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are independent of the state variable x . As a consequence, open-loop and feedback (or Markov) optimal controls coincide, the value (or optimal objective) function is linear in the state x , and the certainty equivalence principle is satisfied.