Parabolic equations without Cauchy boundary condition and control problems for diffusion processes. II.

*(English. Russian original)*Zbl 1031.35504
Differ. Equations 31, No. 8, 1362-1372 (1995); translation from Differ. Uravn. 31, No. 8, 1409-1418 (1995).

[For Part I, cf. [ibid. 30, No. 10, 1606-1617 (1994; Zbl 0855.35050).]

Summary: The present paper continues the study [loc. cit.] of linear parabolic equations with nonstationary coefficients on an infinite time interval in a bounded space domain without Cauchy conditon at the endpoint to the time interval. This condition was replaced by the condition that the \(L_2\)-norm of the solution is finite on the entire infinite time interval. We proved theorems on the solvability of such problems. As a corollary we obtained the solvability conditions for parabolic equations, on a finite time interval \((0,T)\), for which the Cauchy condition \(u(x,T)= 0\) is replaced by a condition of the form \(\kappa u(x,0)= u(x,T)\), where \(\kappa\in [-1,1]\).

In what follows we establish the solvability of a nonlinear (quasilinear) Bellman equation on a finite time interval with a similar boundary condition. The proof is based on the methods of optimal control theory, i.e., we prove that the desired solution is a Bellman function of the control problem for a random diffusion process with nonstationary time-periodic coefficients on the infinite time interval in a bounded space domain.

Summary: The present paper continues the study [loc. cit.] of linear parabolic equations with nonstationary coefficients on an infinite time interval in a bounded space domain without Cauchy conditon at the endpoint to the time interval. This condition was replaced by the condition that the \(L_2\)-norm of the solution is finite on the entire infinite time interval. We proved theorems on the solvability of such problems. As a corollary we obtained the solvability conditions for parabolic equations, on a finite time interval \((0,T)\), for which the Cauchy condition \(u(x,T)= 0\) is replaced by a condition of the form \(\kappa u(x,0)= u(x,T)\), where \(\kappa\in [-1,1]\).

In what follows we establish the solvability of a nonlinear (quasilinear) Bellman equation on a finite time interval with a similar boundary condition. The proof is based on the methods of optimal control theory, i.e., we prove that the desired solution is a Bellman function of the control problem for a random diffusion process with nonstationary time-periodic coefficients on the infinite time interval in a bounded space domain.

##### MSC:

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

35R60 | PDEs with randomness, stochastic partial differential equations |

93E20 | Optimal stochastic control |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |