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AbstractThis paper deals with a class of optimal control problems which arises in advertising models with Volterra Ornstein-Uhlenbeck process representing the product goodwill. Such choice of the model can be regarded as a stochastic modification of the classical Nerlove-Arrow model that allows to incorporate both presence of uncertainty and empirically observed memory effects such as carryover or distributed forgetting. We present an approach to solve such optimal control problems based on an infinite dimensional lift which allows us to recover Markov properties by formulating an optimization problem equivalent to the original one in a Hilbert space. Such technique, however, requires the Volterra kernel from the forward equation to have a representation of a particular form that may be challenging to obtain in practice. We overcome this issue for Hölder continuous kernels by approximating them with Bernstein polynomials, which turn out to enjoy a simple representation of the required type. Then we solve the optimal control problem for the forward process with approximated kernel instead of the original one and study convergence. The approach is illustrated with simulations.

The European Union 2050 climate neutrality goal and the climate crisis require coordinated efforts to reduce energy consumption in all sectors, and mainly in buildings greatly affected by the increasing temperature, with relevant CO(2) emissions due to inefficient end-use technologies. Moreover, the old building stock of most countries requires suited policies to support renovation programs aimed at improving energy performances and optimize energy uses. A toolbox was developed in the framework of the PrioritEE project to provide policy makers and technicians with a wide set of tools to support energy efficiency in Municipal Public Buildings. The toolbox, available for free, was tested in the partners' communities, proving its effectiveness. The paper illustrates its application to the Potenza Municipality case study in which the online calculator DSTool (the core instrument of the toolbox) was utilized to select and prioritize the energy efficiency interventions in public buildings implementable in a three-year action plan in terms of costs, energy savings, CO(2) emissions' reduction and return on investments. The results highlight that improvements in the building envelopes (walls and roofs), heating and lighting and photovoltaic systems allow reducing CO(2) emission approximately 644 t/year and saving about 2050 MWh/year with a total three-year investment of 1,728,823 EUR.