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We use the Valentine transformation to convert a constrained optimal control problem into an unconstrained one and show uniqueness of the value function to the corresponding Hamilton\u2013Jacobi\u2013Bellman (HJB) equation. From there, we show how to approximate the solution of the initial (in)finite-horizon problem with a family of solutions that is<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varGamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-convergent. Optimal solutions are efficiently obtained via a solver based on Pontryagin\u2019s Principle\u00a0(PP). The proposed methodology is demonstrated on the path planning problem using the full nonlinear dynamics of an unmanned aerial vehicle (UAV) and autonomous underwater vehicle (AUV) involving state constraints in 3D environments with obstacles.<\/jats:p>","DOI":"10.1007\/s10957-023-02364-4","type":"journal-article","created":{"date-parts":[[2024,1,9]],"date-time":"2024-01-09T15:03:31Z","timestamp":1704812611000},"page":"437-462","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Application of Hamilton\u2013Jacobi\u2013Bellman Equation\/Pontryagin\u2019s Principle for Constrained Optimal Control"],"prefix":"10.1007","volume":"200","author":[{"given":"Jerome","family":"Weston","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Domagoj","family":"Toli\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ivana","family":"Palunko","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,1,9]]},"reference":[{"key":"2364_CR1","volume-title":"Optimal Control and Viscosity Solutions of Hamilton\u2013Jacobi\u2013Bellman Equations","author":"M Bardi","year":"2008","unstructured":"Bardi, M., Dolcetta, I.C.: Optimal Control and Viscosity Solutions of Hamilton\u2013Jacobi\u2013Bellman Equations. 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