16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Instead, we can use a more round-about procedure. The response of the differential equation for x(o) can be expressed ()=(t,6)(n)+Jc)B(rl(r)dr where qp(t, r) is the transition matrix for the linear system with system matrix A(0. It satisfies the system homogeneous equation a(L, T)=A(oo(t, r) d(z,r)=1 This approach works in this case because we can write down the form of the solution to a set of linear differential equations. We cannot write down the form in the case of nonlinear systems. However, the Ito calculus does apply to roac of the solution to nonlinear differential equations so we cannot use this api nonlinear stochastic differential equations f 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 8 of 8 Instead, we can use a more round-about procedure. The response of the differential equation for x%( )t can be expressed as 0 0 0 () (, ) ( ) (, ) ( ) ( ) t t x t tt xt t B n d =Φ + φ τ τ ττ ∫ %% % where Φ(, ) t τ is the transition matrix for the linear system with system matrix A t( ). It satisfies the system homogeneous equation (, ) () (, ) (, ) d t At t dt I τ τ τ τ Φ =Φ Φ = This approach works in this case because we can write down the form of the solution to a set of linear differential equations. We cannot write down the form of the solution to nonlinear differential equations so we cannot use this approach in the case of nonlinear systems. However, the Ito calculus does apply to nonlinear stochastic differential equations