Fal!2001 16.3122-5 4. Now, since A(t)=P(t)c(t),then (t)=P(t)(t)+P(ti(t) C2 RzCza(t)= A(t) P(t)c(t)=C RzC2a(t)+A X(t)+P( c(t) C2RuCzc(t)+A A(t)+P(t)(Ac(t)-Buruu BIX(t) (C RzC2+P(t)A)a(t)+(A-Pt Buru BnX(t (CZRzCz+ P(t))x(t)+(a-Pt Buru BnP(t)c(t) AP(t)+P(t)A+C2R2C2-P(t)BRuu Bi P(t)a(t) This must be true for arbitrary a(t), so P(t)must satisfy P(t=A P(t)+P(t)A+C2 zC:-P(t BuRuu BiP(t) Which is a matrix differential Riccati Equation The optimal value of P(t) is found by solving this equation back wards in time from tf with P(te)=PtFall 2001 16.31 22—5 4. Now, since λ(t) = P(t)x(t), then λ˙(t) = P˙(t)x(t) + P(t)x˙(t) ⇒ − Cz TRzzCzx(t) − AT λ(t) = −P˙(t)x(t) = Cz TRzzCzx(t) + AT λ(t) + P(t)x˙(t) = Cz TRzzCzx(t) + AT λ(t) + P(t)(Ax(t) − BuR−1 uu Bu T λ(t)) = (Cz TRzzCz + P(t)A)x(t)+(AT − P(t)BuR−1 uu Bu T )λ(t) = (Cz TRzzCz + P(t)A)x(t)+(AT − P(t)BuR−1 uu Bu T )P(t)x(t) = £ ATP(t) + P(t)A + Cz TRzzCz − P(t)BuR−1 uu Bu TP(t) ¤ x(t) • This must be true for arbitrary x(t), so P(t) must satisfy −P˙(t) = ATP(t) + P(t)A + Cz TRzzCz − P(t)BuR−1 uu Bu TP(t) — Which is a matrix differential Riccati Equation. • The optimal value of P(t) is found by solving this equation back￾wards in time from tf with P(tf ) = Ptf