Mild solutions of semilinear elliptic equations in Hilbert spaces

This paper extends the theory of regular solutions ($C^1$ in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of $G$-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into $G$-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

Quickest detection in coupled systems

This work considers the problem of quickest detection of signals in a coupled system of $N$ sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled by general It\^{o} processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. Two main cases are considered; in the first one signal strengths are the same across sensors while in the second one they differ by a constant. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended minimal Kullback-Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time to the first false alarm. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of $N$ CUSUMs is asymptotically optimal as the mean time to the first false alarm increases without bound. In particular, in the case of equal signal strengths across sensors, it is seen that the difference in detection delay of the $N$-CUSUM stopping rule and the unknown optimal stopping scheme tends to a constant related to the number of sensors as the mean time to the first false alarm increases without bound. Alternatively, in the case of unequal signal strengths, it is seen that this difference tends to zero.Comment: 29 pages. SIAM Journal on Control and Optimization, forthcomin

Infinite dimensional parameter identification for stochastic parabolic systems

The infinite dimensional parameter estimation for stochastic heat diffusion equations is considered using the method of sieves. The consistency property is also studied for the long run data

Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems

We introduce a new method, allowing to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.Comment: 37 pages, 11 figure

Stochastic Control Problems with Unbounded Control Operators: solutions through generalized derivatives

This paper deals with a family of stochastic control problems in Hilbert spaces which arises in typical applications (such as boundary control and control of delay equations with delay in the control) and for which is difficult to apply the dynamic programming approach due to the unboudedness of the control operator and to the lack of regularity of the underlying transition semigroup. We introduce a specific concept of partial derivative, designed for this situation, and we develop a method to prove that the associated HJB equation has a solution with enough regularity to find optimal controls in feedback form