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We study the optimal control of path-dependent piecewise deterministic processes. An appropriate dynamic programming principle is established. We prove that the associated value function is the unique minimax solution of the corresponding non-local path-dependent Hamilton-Jacobi-Bellman equation. This is the first well-posedness result for nonsmooth solutions of fully nonlinear non-local path-dependent partial differential equations. 

We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton-Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with respect to time. We apply our results to optimal control problems of (delay) functional differential equations with cost functionals that have discount factors and with time-measurable data. Our main results are also crucial for our companion paper Bandini and Keller [arXiv preprint arXiv:2408.02147 (2024)], where non-local path-dependent Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied. 

We establish necessary and sufficient conditions for viability of evolution inclusions with locally monotone operators in the sense of Liu and Röckner [J. Funct. Anal., 259 (2010), pp. 2902-2922]. This allows us to prove wellposedness of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations associated to the optimal control of evolution inclusions. Thereby, we generalize results in Bayraktar and Keller [J. Funct. Anal., 275 (2018), pp. 2096-2161] on Hamilton-Jacobi equations in infinite dimensions with monotone operators in several ways. First, we permit locally monotone operators. This extends the applicability of our theory to a wider class of equations such as Burgers' equations, reaction-diffusion equations, and 2D Navier-Stokes equations. Second, our results apply to optimal control problems with state constraints. Third, we have uniqueness of viscosity solutions. Our results on viability and lower semicontinuous solutions are new even in the case of monotone operators. 

In commercial large-scale aquaria, controlling levels of nitrogenous compounds is essential for macrofauna health. Naturally occurring bacteria are capable of transforming toxic nitrogen species into their more benign counterparts and play important roles in maintaining aquaria health. Nitrification, the microbially-mediated transformation of ammonium and nitrite to nitrate, is a common and encouraged process for management of both commercial and home aquaria. A potentially competing microbial process that transforms ammonium and nitrite to dinitrogen gas (anaerobic ammonium oxidation [anammox]) is mediated by some bacteria within the phylum Planctomycetes. Anammox has been harnessed for nitrogen removal during wastewater treatment, as the nitrogenous end product is released into the atmosphere rather than in aqueous discharge. Whether anammox bacteria could be similarly utilized in commercial aquaria is an open question. As a first step in assessing the viability of this practice, we (i) characterized microbial communities from water and sand filtration systems for four habitats at the Tennessee Aquarium and (ii) examined the abundance and anammox potential of Planctomycetes using culture-independent approaches. 16S rRNA gene amplicon sequencing revealed distinct, yet stable, microbial communities and the presence of Planctomycetes (~1–15% of library reads) in all sampled habitats. Preliminary metagenomic analyses identified the genetic potential for multiple complete nitrogen metabolism pathways. However, no known genes diagnostic for the anammox reaction were found in this survey. To better understand the diversity of this group of bacteria in these systems, a targeted Planctomycete-specific 16S rRNA gene-based PCR approach was used. This effort recovered amplicons that share <95% 16S rRNA gene sequence identity to previously characterized Planctomycetes, suggesting novel strains within this phylum reside within aquaria. 

AGAMOUS-like 15 (AGL15) is a member of the MADS domain family of transcription factors (TFs) that can directly induce and repress target gene expression, and for which promotion of somatic embryogenesis (SE) is positively correlated with accumulation. An ethylene-responsive element binding factor-associated amphiphilic repression (EAR) motif of form LxLxL within the carboxyl-terminal domain of AGL15 was shown to be involved in repression of gene expression. Here, we examine whether AGL15′s ability to repress gene expression is needed to promote SE. While a form of AGL15 where the LxLxL is changed to AxAxA can still promote SE, another form with a strong transcriptional activator at the carboxy-terminal end, does not promote SE and, in fact, is detrimental to SE development. Select target genes were examined for response to the different forms of AGL15. 

Abstract We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle.