Journal of Nonlinear and Variational Analysis

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DOI: 10.23952/jnva.11.2027.1.09

Volume 11, Issue 1, 1 February 2027, Pages 187-213

 

Abstract. We study the mathematical foundations of the least-squares parameter identification problem for the smart thermostat thermal model introduced by Arias et al. (Appl. Math. Model. 2022). This model depends on four parameters: heat capacity, thermal resistance, effective boiler power, and a solar radiation factor. We generalize previous results on existence, stability, and differentiability from continuous to essentially bounded data by considering Carathéodory-type solutions. Furthermore, we rigorously justify the need for regularization to avoid spurious multiple solutions. To address this, we establish a reformulation in terms of three parameters that retains physical interpretability, and we prove existence, stability, and differentiability results for it. In particular, we obtain an explicit characterization of the Hessian of the reformulated objective map. As a consequence, we prove its strict convexity over a range of regularization parameters, thereby guaranteeing uniqueness for the identification problem. We rigorously show how, under a compatibility condition between both models, this result leads to uniqueness, or improved approximation when combined with a priori information, for the general four-parameter model. Finally, following a first-optimize-then-discretize approach, numerical experiments with synthetic data and with semisynthetic data based on real measurements confirm our theoretical findings and demonstrate the superior performance of the reformulated three-parameter model.

 

How to Cite this Article:
J. Arias, A.A. Khan, M. Sama, Convex reformulation of a 1R1C smart thermostat thermal model, J. Nonlinear Var. Anal. 11 (2027), 187-213.