Does consumption of amino acid supplements improve human executive function and self-regulation in ADHD cases? Based on my understanding, executive function primarily takes place in the frontal lobes. Inzlicht and Gutsell found that exhaustion in mental tasks shows itself in decreased activity in the anterior cingulate cortex. The main neurotransmitter in that region, and in the frontal lobes in general, seems to be glutamate.  L-Theanine inhibits reuptake of glutamate via EAAT2. However, when a person consumes a L-Theanine supplement, it might not be necessarily the case that the human body will update the neurotransmitter distribution beyond the blood-brain barrier in a tantamount fashion. 

L-tyrosine, like amphetamines (prescribed for ADHD), seems to increase dopamine by reuptake inhibition. I have similar concerns to the above (i.e., it might not be necessarily the case that the human body will update the neurotransmitter distribution beyond the blood-brain barrier in a tantamount fashion).  

In general, there tends to be a lot of quackery and a lot of p-hacking with small samples when it comes to uses of L-Theanine supplements and natural remedies on health. This might be especially the case in neuroscience, where a lot of treatment effects are hard to trace.
There's a recent negative study: Bergwerff et al. (2016) "No Tryptophan, Tyrosine and Phenylalanine Abnormalities in Children with Attention-Deficit/Hyperactivity Disorder": 


  This study is the first to explore AAA metabolism in children with ADHD using a well-defined and relatively large sample. We found that AAA deficiencies are not related to ADHD. The results do not support treatment with AAA supplements in children with ADHD. 


Also reviews prior work on defficiency:


  Thus far, five case-control studies have been published on AAA blood concentrations in individuals with ADHD [21–25]. Three studies, of which two describing the same sample [21, 22], reported lower plasma concentrations of tryptophan, tyrosine and phenylalanine in ADHD [21–23]. The other two studies, however, showed increased concentrations of free tryptophan in ADHD [24] and a trend towards increased serum concentrations of tryptophan in children with ADHD [25]. All five studies are limited by non-standardized assessments of ADHD and small sample sizes (ranging from N = 12 to N = 48), and therefore further research into the availability of AAAs in ADHD is warranted. [thus their own study.]


After their negative finding Bergwerff et al. do mention however that:


  We found a relatively high odds ratio of 2.2 for being diagnosed with ADHD when having low phenylalanine concentrations, but this result just escaped conventional levels of significance. Therefore, our study does not definitively rule out that low phenylalanine concentrations are present in (a subgroup of) children with ADHD. For instance, it might be that only children with severe deficiencies in executive functioning have decreased phenylalanine concentrations, as an altered dopamine functioning in the prefrontal cortex and the striatum is thought to impair executive functions, including sustained attention and interference control in ADHD [8, 67]


And they also reviewed treatments with supplementation:


  The hypothesis that AAA concentrations are related to ADHD symptoms is the basis for a number of depletion and supplementation studies. Depletion of dietary tryptophan was found to impair sustained attention in adults with ADHD [30], and to weaken behavioural inhibition in hostile children with ADHD [31]. Supplementation with tryptophan, on the other hand, resulted in a decrease of ADHD symptoms in children with ADHD [32]. Tyrosine supplementation decreased ADHD symptoms in adults with ADHD [33], but showed no effects on behavioural functioning in children with ADHD [32]. Phenylalanine supplementation in adults with ADHD caused a decrease of restlessness and an increase on the ability to concentrate at trend level [34], but in children no effects were reported for phenylalanine supplementation on ADHD symptoms [35]. However, also these depletion and supplementation studies are limited by non-standardized assessments of ADHD and small sample sizes (ranging from N = 10 to N = 20), as well as the lack of control groups, hampering conclusions regarding the relation between AAAs and ADHD. Therefore, there is a need of further research to support the hypothesis that AAA concentrations may contribute to the expression of ADHD symptoms.


So evidence for AAA supplementation effectiveness for treating ADHD is insofar of low quality; and there are contradictory findings (to put it mildly) on the actual AAA deficiencies in ADHD.

And regarding L-theanine, there's a study, Lyon et al. (2011) on sleep quality in ADHD. Which is probably the way they "salvaged" not observing much else in terms of improvements. The (2006) registration info for that trial had loftier goals to "examine the effects of L-theanine (an amino acid found in green tea) on the behavior, cognitive performance and sleep quality of boys with ADHD."

I tried to find a more systematic (or even meta-analytic) review of supplements in ADHD, but not much with that. It looks like most research on ADHD and diet is on restrictions (i.e. what not to eat), and even that area is full of controversy.